We parked in the "public " lot and were run out by the cops eventually. As long as you don't pop the top you may be ok, but with a crew onboard you might find it difficult. You will not be able to park on side streets and camp, as the cops are really fastidious about running car campers off the island. They will tell you to go out to Stock Island and camp (NIMBY!)...which you may not want to do because it has some seedy areas. Emergency camping is to be found out in the Kmart parking lot near up near the bridge on KW.
I would HIGHLY recommend Lazy Lakes Campground out on Sugarloaf key...12 mm I think. Go to www.clydeskeywest.com
 and click on "camping" then "key west" (not middle keys, or upper keys). There you'll find the phone number of many other spots including Lazy lakes.
We loved LL because it startes out as the first hippie camp spot in the 60's and developed into a home-y little budget campground over the years. There are showers, laundry, pool, and a game room. Also it is very pet friendly, with a dog walk area (complete with pooper-scoopers) encircling the "lake". They offer kyaks and canoes for free in the "lake" (a large pond) as well. We paid $36 a night at the height of the season last year (New Years week) for a spot with water hook-up. Any more info you might need, email me.
Oh yeah...see you in KW this winter!

Tracey