The definition of a 'public road' under The Road Traffic Act 1988 is different from the definition used by Vehicle Excise and Registration Act 1994 [which governs excise licences] Hence, although offences under The RTA 1988 may well be committed in car parks, petrol forecourts etc. the offences relating to vehicle excise licences fall into a different definition viz.
Vehicle Excise and Registration Act 1994 (c. 22)
“public road”—
(a) in England and Wales and Northern Ireland, means a road which is repairable at the public expense, and
(b) in Scotland, has the same meaning as in the [1984 c. 54.] Roads (Scotland) Act 1984
From the DVLA themselves, the advice on untaxed vehicles states;
If you see a vehicle on the public road (car parks, driveways or housing association parking spaces don’t apply) with an out of date tax disc, you can report it online, by telephone or by post. You don’t have to give your name. Your report will be investigated and enforcement action taken, if needed.
In London, there still exist police traffic wardens as well as local council wardens. Police traffic wardens can issue notices and have access via radio to The Police National Computer just the same as police officers.
Police traffic wardens and police officers have always been able to report untaxed vehicles to The DVLA. In recent years, police traffic wardens have also been able to issue penalty notices for non-diplay of VEL [Before that only police officers could issue such a penalty notice]
I am not aware that the power to issue tickets for non-display of VEL has been legalised for local council wardens albeit, just like joe public, they can report untaxed vehicle to The DVLA.
Therefore, I would suggest that the offence under Vehicle Excise and Registration Act 1994 is not committed in a car park such as the OP parked in and, in any event, even if it was the local council warden could not issue a ticket for it.
I think it far more likely that the car park terms and conditions enshrine a need to display a valid VEL and any breach of these car park terms and conditions would enable the local council warden to issue a ticket for contravening the car park terms [as opposed to the Vehicle Excise and Registration Act 1994 offence]