If a company owns a piece of land, and has paid out to turn it into a parking lot, and maintain and patrol the area, then they have every right to charge people for parking there. If someone parks illegally, then the company should be able to recoup the costs by issuing a parking ticket.
While you're on private property, the owner of the property has the right to give tickets to people on that property. However, they can't just simply walk through town and start putting tickets on cars. Imagine for a second a busy parking garage during a convention weekend. If some jerk parks using three spaces, the company clearly has a right to charge him for the lost income.
Universities, stores, and industries that are responsible for maintaining their own parking areas have the right to establish rules for parking on their property. They can have their own security force that enforces those rules by booting vehicles and issuing parking tickets. The private company has a right to determine how parking is done on their property.
No, private companies should not be able to fine citizens. The sticking points are enforcement if the car owner refuses to pay, and the rights of the wrongfully fined car owner. Both instances require a level of legal authority private companies don't have, nor should have. The potential for abuse is just too great.