I would add, try a private campground. Florida is loaded with these places and most have plenty of space. Maybe, you could arrange to stay at the campground at the beginning and end of your trip while they keep your car parked out of the way for a small fee.

The campgrounds range from small mom and pop places to large national chains. Some are resorts, so try several to see who will give you the best deal.

As a side note, a few years ago my wife and I were looking for a place to store our motorhome, so I went through the yellow pages calling campgrounds. Almost all were receptive. Of course storing RVs is part of what they do. We then went from campground to campground to inspect the storage facilities, security etc.. One campground was surrounded by a tall board on board fence and had an Africian safari theme. Security was heavy at the front gate. After we got inside the gate we found out why. It was a nudist camp. It was kinda weird at first, being shown around by a naked man and woman. The place was crowded too. We got invited to a round of naked golf. We declined. Anyway, I voted to keep the motorhome there, you know, reasonable prices and good security and all. In the end my wife veto'd the idea and we store Homer the motorhome on a farm.