You'd almost be better off seeking out a large marina parking lot and just pulling in like you're parking there while "working" on your boat. Or maybe stopping in a larger town like Marathon and parking on a side street, possibly asking a homeowner if it's ok to park on the street near their house over night. During the day just go down to Key West and find some street parking 4-5 blocks from Duval Street or maybe a beach parking lot during the day. Left or right off Truman Avenue, a main street heading down to Duval Street, has free street parking. We camped up in Marathon and drove down to KW and parked in the area I mentioned. Walked back to our vehicle in the evening. No problem.
Overnights may still be a problem, but I don't think the local law enforcement will allow you to just pull over along the Hwy 1 for the night. Look like you "belong" there.
