The actor, writer and producer Rashida Jones suggested we meet at a private members' club near her home in West Hollywood, which seemed like a fine idea at the time. But, as I wait for her, I worry that there must be a dress code that I didn't know about, because every single woman but me is wearing skinny designer jeans, metallic heels, a sexy top, a full face of makeup and an enormous designer handbag, at 3pm on a sunny Saturday. After a few long minutes in which I shift self-consciously around in my floral cotton dress, Jones comes clattering towards me, barefaced and dressed, in her words, "like a mime": in flats, wide-legged cropped black trousers and a Breton top. She looks – and I mean this in the non-infantilising sense – adorable. She also looks a little out of place here, but her reason for suggesting the venue becomes quickly apparent.

"They have the best chocolate chip cookies," she says, ordering a plateful as soon as she sits down.

I tell her I was worried the club had a dress code. "Oh God, that whole LA Barbie doll look, right?" she says. "It's weird that everybody wants to look like everyone else. I love what you can do with fashion, but that look is just not my nature. I like conservative dressing. I don't like to dress to tell people that they want to have sex with me."

While I was waiting for her, I say, men kept approaching the two women sitting near us, who are wearing extremely sexy versions of this LA uniform, and one chat-up line was, "Can I just charge my iPhone here?"

"Yeah, more like, 'Can I just charge my penis over here?'" Jones cackles.

In Los Angeles, most celebrity offspring ride a merry-go-round of reality TV, sex tapes, rehab and mug shots, but Jones, 37, the daughter of musician Quincy Jones and actor Peggy Lipton, has done things a little differently. A Harvard graduate (religion and philosophy) who once planned on being a judge or the president, she is most famous as a comedic actor, particularly for her role as Ann Perkins in Parks And Recreation. Made by the same team that brought The Office to America, it bears a resemblance to its predecessor, only with a more female focus; the show also stars the brilliant Amy Poehler.

Before Parks And Recreation, Jones spent years appearing on TV shows such as Freaks And Geeks and Boston Public; more recently, there have been feature films – I Love You, Man with Paul Rudd, and The Social Network. Whatever the role, Jones brings both sweetness and realism, with a touch of goofiness (she is especially good at heartbreak), and manages to rise above the, to put it bluntly, somewhat generic roles into which she has been pushed. In the decidedly bro-centric I Love You, Man, she played Rudd's fiancee, a woman who doesn't understand his close male friendship with Sidney (Jason Segel). But instead of simply playing the nagging girlfriend, Jones made the character sympathetic, conveying her understandable bafflement at why her fiance would discuss their oral sex habits with his new friend. Her most recent role, in the salsa dancing British romcom Cuban Fury (she plays Nick Frost's love interest), doesn't really do her justice. But she talks excitedly about how much she loved the classic British TV show Spaced, in which Frost appeared. "I'm obsessed with Nick Frost – he's so dreamy," she says, and then adds, less controversially, "I also wanted to learn how to salsa."

"Weird, right? A Hollywood kid who doesn't seem like a total asshole?" the feminist website Jezebel wrote about Jones last year. And it's true. She has all the credentials to be a total cliche – she even went to the same school as Paris Hilton and the Kardashians – yet she's thoughtful, unselfconscious and funny. I have interviewed some of those "Hollywood kids". One – so thin I could nearly see through her – binge-ate throughout the interview; another informed me that he "refused to be interviewed by the work-experience girl" (I was 27). Jones, by contrast, extends our time together twice over and sends me off with a box of cookies.

Did she consciously react against the brat cliche?

"It was never a reaction, really. I'm obsessed with the nature/nurture argument, and the older I get, I think nature is paramount to everything. I am who I am, and I wanted to go to Harvard when I was four years old, and I still wanted to go when I was 18, and so I went. I was never going to be like, 'You know what my last name is, right?'"

As a teenager, she says, she was "a chubby nerd. Nobody was trying to have sex with me, so I had to find other things, like reading and being good at school."

Your identity as a teenager shapes you for ever, I say.

"That is so it. Every time I look at a photo of myself as a teenager, I think, this is who I feel like." She shows me a photo she keeps on her phone of an overweight, solemn-faced girl who looks vaguely like Rashida Jones. "There I am. So uncomfortable, so sad, so much food… On family vacations, I used to wake up before everyone else and make a sourdough bread sandwich with butter, eggs, bacon and cheese, and then go back to bed and, like, one hour later be, 'So what's for breakfast?'" She laughs. "So sad."

With parents like hers, I say, it must have been awful being a chubby nerd. Her father was producing Michael Jackson's Off The Wall and Thriller albums when Rashida was a child; her mother was a famous beauty, now best known as the blonde who runs the Double R Diner in Twin Peaks. "I definitely felt different as a kid. But my parents are hippies in the right way, in that they supported anything I was interested in. So when I would talk about being interested in the law as a kid, they would be like, 'That's so cute! She has her own thing!' The big twist in events is that, after college, I realised I wanted to be an actor, because that wasn't what I had thought I wanted from my life."

Now Jones is stepping away from acting and, tragically for fans of Parks And Recreation, this means she is leaving the show in the middle of the sixth series.

A few years ago, she and her best friend, Will McCormack, wrote the lovely and bittersweet romcom Celeste And Jesse Forever, in which she starred alongside Andy Samberg. The critic Roger Ebert praised Jones's acting and screenplay, saying that she and McCormack "write dialogue and create supporting characters who don't seem like airheaded plot puppets, the way so many romcom characters do. There's a certain respect for their feelings." Now they are developing six TV shows, both comedy and drama, featuring different kinds of female protagonists. "We're trying to inundate the market with women characters and create choices based on quality and not on stereotypes," Jones says.

It feels redundant to ask, but has she struggled to find interesting roles? She nods emphatically. "I realised I was in this small space with all these super-talented women vying for one part in one movie, and it was always the shitty part in that year's one good movie: someone's wife, or someone's shrew girlfriend. So that has definitely played a part in my decision to act a little less and create a little more, because I want to add more to that conversation about what it takes to be a woman."

She has already started doing just that. She has tended to be cast as bland and inoffensive, but in Celeste And Jesse Forever she created for herself, if not an unlikable character, then certainly a complicated one: a woman who suddenly realises she doesn't know what she wants from life and lashes out at those she loves most.

I tell her that one of my favourite things about Parks And Recreation is the friendship between Jones's character, Ann, and Amy Poehler's character, Leslie. It's a rare depiction of female friendship that isn't predicated on the two women talking about men, or fighting and then making up. "It was really important to Amy that it be like that, and I think it helped that we were friends before, so there's a lot of us in Ann and Leslie."

Jones is still close to her girlfriends from school, but she also has a tight, supportive group of female friends in the industry. Whereas other Hollywood kids tend to stick together, Jones branched out to find a more interesting set of smart female comedians, many ex-Saturday Night Live performers, including Poehler and Tina Fey. If in her 20s she wanted to hang out with "the cool crowd, because I wanted to be cool, now I look for people with good hearts". When the legendary 80s film-maker John Hughes died in 2009, Jones spent the day swapping favourite quotes from Hughes's films with Poehler, Tina Fey and actor Maya Rudolph, in what sounds like the best email chain ever. "I definitely like women in comedy," she says, "but I feel a little on the outside because I generally play the straight guy. So I'm just like a fan with them."

Last year, Jones wrote an article for Glamour magazine, despairing about the pornification of pop culture. "Stripper poles, G-strings, boobs and a lot of tongue action [are] all now normal accessories for mainstream pop stars," she wrote. She also addressed the accusation that she was "slut-shaming" by criticising women's overly sexualised clothes: "I consider myself a feminist… There is a difference, a key one, between shaming and holding someone accountable."

"I'm just asking people to take a breath and talk about it," she tells me now. "I also wanted to say there's more than one way to be a woman and be sexy – like, you're a really great dancer, or you're really fucking smart."

She recently signed a deal to write a monthly relationships column for Glamour. "I'm totally obsessed with relationships, female friendships, romantic relationships, all of it. It just fascinates me, thinking about what works, what doesn't. I'm single, you know, so how does that make me different from my friends with kids? Do I want kids?" she says.

Jones has had several long-term relationships, including with Tobey Maguire and DJ Mark Ronson. Did she have an idealised image about relationships when she was younger? "I had the full princess fantasy: the white horse, the whole being saved from my life, which is ridiculous. What do I want to be saved from? My life's great! But it's just this weird thing that's been hammered into my head culturally: that's the only way to succeed, that's the only thing that counts for a woman. I'm happy, but the fact that I'm not married and don't have kids – it's taken me a long time to get to a place where I actually am OK with that, where I actually don't feel like I'm some sort of loser."

When it was announced that Jones would be writing a relationships column, Jezebel – the blog that had praised her just months before – sneered, "I'd rather someone in a successful relationship tell me about relationships", apparently oblivious to the irony of a self-styled feminist website criticising a woman for being single.

"I saw that, and I'm old enough to say, just fuck off," she says, her eyebrows diving down at the memory. "But to suggest that the way to be a feminist is to snark about other women is such a dangerous example to set."

Jones was born in 1976, the second daughter of a black man and a white Jewish woman, who married in 1974 when mixed-race couples were a definite anomaly, certainly in the US; they divorced when Rashida was 10. In a 1993 interview, the late Tupac Shakur attacked Quincy Jones for having children with a white woman. Rashida, then only 17, wrote a furious letter to the magazine. She reels in embarrassment when I mention this. "Oh God, I can be self-righteous to a fault, but I think it says a lot about my parents that I felt able to speak up like that, you know."

The twist to this tale is that a few years later, Shakur met Jones's older sister Kidada, and apologised for what he had said. The pair later got engaged. "That was a nice full circle, and it was a good lesson for me, because I knew in my heart of hearts that I'd never forgive anyone for talking like that about my family, and I was wrong."

It has taken Jones some time, but she has reached a point where she can talk proudly about her parents' achievements without worrying about people thinking she owes her career to them.

"I would never say 'poor me', but there definitely was a lot of pre-judgment about how I got my job in the beginning, when the truth is I was just another actress getting knocked back at auditions for years. I do have a hang-up about it sometimes," she says, and then changes her mind. "No, I just don't care any more." These days she is just happy no longer "having to fight for the shitty role. I feel things are changing, and there are real female comedy stars now – and that's exciting, isn't it?" she says, eating her last cookie, a mime among Barbie dolls.