The AMA has a flying site coordinator who's supposed to help clubs out with that sort of thing. I'd suggest you start with him -- hopefully he'll be full of helpful suggestions. I can't say how good the program is -- according to the AMA it's the best thing since sliced bread, but I haven't heard about it outside of the AMA. Here's a page on the AMA site:
http://www.modelaircraft.org/membership/clubs/fsap.aspx.
Aside from that, the obvious: be polite ("would it be possible", not "the world owes us"), be sensitive to objections (they built those parking lots to park cars, which you'd be displacing -- you may need to agree to not use them on certain days, or pack up and move when other lots are full, etc.), think of what your club could do to give back, be ready to point out advantages to the community, etc.. For goodness sake, if the park director's hands are tied for some stupid reason don't say "well that's stupid!". It may help to be flexible, too -- if the parking lots can't be used for one reason or another, but if there's a suitable open space available, consider whether you want to ask about it, and either fly off of grass or see if you can build a circle (at your club's expense, of course) in that space.
My RC club (where a handful of us have been flying CL for the last few years) is at an Oregon State park. We've been there for just about ever so I don't know how we got in originally, but the deal is that we maintain our little portion of the park (we're behind a locked gate) in return for having it. The park likes us because we're not demanding and we keep our part of the bargain.
As far as keeping it once you get it, in our case we do try to maintain a helpful attitude: at a meeting a couple of years ago of all the special-use groups at the park the park asked what people thought could be done to improve the relationship. The bicycle folks asked for covered bike storage, the horse people asked for more trails, the frisbee golf people asked for designated parking, the Civil War reenactment folks had some request or another -- our representative said "is there anything we can do to make your lives easier?".