It is fair to require people to pay to park their cars in public places. Parking spaces are in high demand. If someone has enough money to buy a car and drive it into an expensive downtown area, they should have no trouble paying to park it for a few hours.
Yes, people should have to pay to park their car if the person who owns the parking space wants them to pay. Private property allows people who own real property to charge other people to use it. Also, charging for parking reduces demand, and allocates the parking spaces to those most willing to pay for it.