Are parking lots where you pay for a reserved parking spot private or public?

Is parking lot where you pay monthly to park your car private? Everyone can pay and park, but you cannot spontaneously come and park your car for a few hours, usually you’d need some kind of a contract.

As I understand parking lots where you can leave your car for a few hours if there is a free spot are public, but in this case it’s access=private, am I correct?

access=permit sounds a lot like the other suggested uses of that tag. It means ˇYou need to arrange something beforehand¨

Maybe access=permit would make sense? Maybe along with fee=yes?