It is actually possible to park on the street in Rome overnight for free. Helpful if you can read enough Italian to avoid the morning tow away zones. If you have an out of Italy registration, some have been known to ignore the parking tickets (which the law abiding Italians, of course, always pay). I don't think it would be wise to leave a fancy, newish rental on the street though.

If you are renting a car why not turn it into the rental company when you arrive and rent another when you want to leave?