By the rental agreement I was the only one authorized to drive the car, so I don't know if I can claim that the driver is unknown by me ...
You are not claiming that the driver is unknown to you. The Hire Company does not know you were the driver. They cannot prove you were the person who drove the car onto the private land. All they can do is provide the PPC evidence you were the Hirer at the time of the alleged breach of contract.
The PPC does not know you were the driver. They are not allowed to presume nor infer that you were the driver just because you happen to be the Hirer. The driver and the Hirer/keeper are two separate entities in civil law. There is plenty of persuasive precedent to validate that point in law.
There is no legal obligation for the Hirer to identify the driver in civil law. Unlike in criminal law, where the Hirer could be required to identify the driver, it is not the case here.
Anyway, this is all academic as the Parking Charge Notices (PCN) have been cancelled. All this argument about who is and who is not liable for the alleged breach of contract does not matter anymore. It is only an argument if it was being appealed or if the Hirer were being sued for the alleged debt in county court.
This is now a contractual dispute over the hire agreement. It is betond our remit except for the fact that we fancy ourselves to be somewhat knowledgable in matters of contract law without the required qualifications of being actual lawyers.
I’m pretty sure that Enterprise Car Hire is a member of the BVRLA and that ATA has guidelines for its members that they are expected to abide by. One of their guidelines is about differentiating between penalties and fines for traffic offences which are criminal matters (although many of the offences are de-criminalised) which, if they ever got to court would be dealt with by a magistrate and any fines imposed would go to the public purse and Parking Charges which are simply civil matters involving contract law and are nothing to do with authorities of any kind as they are simply invoices issued for alleged breach of contract by unregulated private companies.
Unless the hire agreement differentiates between “offences” that incur “penalties” and/or “fines” or “charges” from unregulated private companies, then you have an argument as to whether the agreement allows the hire company to charge you an admin fee.
So, the issue about not revealing the drivers identity is now moot in this case. Good luck with your dispute over the admin charge but feel free to let us know how you get on.