Who operates it is irrelevant; it's about who owns it.  

The state or anyone else owns anything you leave on their land, and since in this scenario you didn't pay them anything, if they decide to keep it, you have no natural right to get it back without forcefully taking it back from their land.  

If you did pay them, then forceful retrieval is justified (but infeasible against the state); if not, you'd simply be the asshole in the situation.  Statists would argue that the latter is actually okay, and the state is always available to help you get your stuff back, in case the landowner gets grabby with the things on their land (ie. exercises their right to liberty) 🤣 

 Do you think paying taxes counts as paying for public parking? In my eyes, anything owned and operated by the state should be free to me as long as I pay my taxes. 

 It's up to the landowner (municipality, homeowner, business owner) whether the tax/transaction contract includes "vehicle storage in designated areas"

Paying taxes doesn't automatically give a blank check right to everything the government owns; it's (ostensibly 🤣) a social contract with terms that both sides understand, agree, and adhere to. 

 When considering the reality of things, I agree. In my hopeful middle-ground utopia, taxes would be the only source of income a government has, and anything that falls outside of that budget wouldn't be paid for by the public sector.