Who is filming the Pawnee Parks and Recreation department?

Just a documentary crew. The makers were inspired by The Office:

You don’t choose mockumentary randomly as a style to shoot a show in—you choose it because it serves the themes and the ideas of the show. And certainly with The Office — and this came right from Ricky Gervais and Stephen Merchant — the idea was that we were going to be a fly on the wall, we’re going to make it look like a real documentary of the middling lives of these average people in this average business in this average town.

Such documentaries are regularly appearing on British TV (about airports, cruise ships, phone centers etc.) and thus the format is a useful one to use as a hook for a comedy. Especially considering the lead character:

The main character of that show, both in the British and American versions, the whole point was that he thought he was hilarious, and he believed deeply that everybody loved him, and so the mockumentary style was great for that, because it showed him acting for the cameras, trying to be this huge, funny clown. And then you got to whip over and see the less-than-enthusiastic reactions of the people for whom he was ostensibly performing. That style fit that show perfectly, and I don’t think you could have done The Office, British or American, without the mockumentary style, without the tiny glances to camera, and without the main character who is performing all the time.

However, this turned out to be not as applicable to P&R:

In our show, we realized early on that Leslie is not performing for anyone. Leslie is completely authentic through and through, she doesn’t care what people think of her, necessarily, or whether she comes off as cool, or any of the stuff Michael Scott or David Brent cared about. That means it is not as vital that we stick to that style, because it is not as vital to the theme of the show.

And thus the style was partially abandoned by season two. However, the makers did keep the bits they liked:

We still love it, because it allows us to get exposition out in funny, brief ways — you know, people can just tell the cameras what is going on, which is a very excellent tool. It’s a tool that, if you don’t have it, you end up with a lot of conversations where people speak unnaturally, where they say things like, “Bill, you’re my best friend, and we’ve known each other for 25 years” — things no one would ever say to each other. We don’t have to do any of that, because we have the device of people explaining to the cameras what’s happening that day. I still love that, and I can’t imagine doing TV without it.

Additionally, while Leslie might not "play it up" for the cameras, April's sarcastic glances to camera (a la Jim from the Office) are an earnest and crucial piece of comedy within the show. As well, P&R utilizes the doc style to capture private character moments in a fly on the wall way -- from Ron secretly checking that no one stole his secret bacon stash to Ben unabashedly working on his claymation project -- things that a Cinema Verite style of filming really lends itself to.

I think the best thing Parks did was create characters with such strong ethos's and senses of self that it feels natural when they share their opinions in talking head interviews.