I think the first point to establish is the fact that YourParkingSpace may have some liability, at least under the Consumer Rights Act 2015 (CRA).
To help us establish this, it would be useful to look to the YourParkingSpace terms and conditions: https://www.yourparkingspace.co.uk/company/parking-contract (https://www.yourparkingspace.co.uk/company/parking-contract)
This suggests that a contract is formed at the time of booking between the owner of the space and the person booking. I wonder here if an argument could reasonably be made that the driver had a contract to park, between him and the owner of the space, and therefore primacy of contract applies? If he already has a contract to park, can he be deemed to have entered a separate contract to park with ParkingEye?
YourParkingSpace state under their terms that the driver must "use the Parking Space in line with any other requirements specified in the booking confirmation". I'd suggest that this is an unfair term, as it seems to be very close to one of the examples of terms that may be regarded as unfair, from Part 1 of Schedule 2 (https://www.legislation.gov.uk/ukpga/2015/15/schedule/2) of the Consumer Rights Act:
"A term which has the object or effect of irrevocably binding the consumer to terms with which the consumer has had no real opportunity of becoming acquainted before the conclusion of the contract."
After all, it seems the requirement to register one's VRM on a terminal was only communicated to the OP after the booking had been made.