You can run a private car park on private land, no question. what you can't do is issue a penalty or a fine for illegal parking, including over staying or not paying, because there hasn't been any offense, and even if there was the private firm isn't entitled to issue penalty charges. They can raise an invoice and claim that there was a contract, but only for the parking, not for any punitive penalty.
Often the Yellow notice on the windscreen looks like a Penalty Notice but is in effect only a dishonestly inflated bill. It's a disgrace really, it's like obtaining money by impersonating a police officer.
Tim w