You have to understand that you are dealing with a vexatious firm of unscrupulous ex-clampers. They have no idea who was driving and you must not reveal that information to them.
As the Registered Keeper, you are under no legal obligation to revel the drivers identity to an unregulated private parking company. VCS are not allowed to assume or infer that the Keeper must also be the driver.
As the alleged contravention took place on airport land which is under statutory control, they cannot rely on PoFA to transfer liability from the unknown driver to you, the Keeper.
Also, they are relying on contract law to say that the driver entered a contract with them. However, their signs are unable to form a contract because they are forbidding.
So, you must write to the LBA airport management and complain vigorously about the actions of their agent, VCS, in issuing you, the Keeper, a Parking Charge Notice (PCN). As they did not form a contract with he driver, they have breached the KADOE contract with the DVLA and obtained your data unlawfully. That is a breach of your GDPR and unless the PCN is cancelled you will hold them jointly and severally liable for the action of their agent VCS and reserve the right to sue for compensation under the Data Protection Act 2018 (DPA).
I suggest you send that complaint to the LBA CEO. Just remember, you are complaining as the Keeper who has received the PCN. No "I did this or that". You refer only to the driver in the third person "The driver did this or that".
Before the deadline to appeal expires, if you have not heard back from the Airport, you send the follwingto VCS as your appeal (only appeal as the Keeper):
I am the registered keeper. VCS cannot hold a registered keeper liable for any alleged contravention on land that is under statutory control. As a matter of fact and law, VCS will be well aware that they cannot use the PoFA provisions because Leeds Bradford Airport (LBS) is not 'relevant land'.
If LBA wanted to hold owners or keepers liable under Airport Bylaws, that would be within the landowner's gift and another matter entirely. However, not only is that not pleaded, it is also not legally possible because VCS is not the Airport owner and your 'parking charge' is not and never attempts to be a penalty. It is created for VCS's own profit (as opposed to a bylaws penalty that goes to the public purse) and VCS has relied on contract law allegations of breach against the driver only.
The registered keeper cannot be presumed or inferred to have been the driver, nor pursued under some twisted interpretation of the law of agency. Your NTK can only hold the driver liable. VCS have no hope at IAS, so you are urged to save us both a complete waste of time and cancel the PCN.
It will be rejected but you will have put then on notice that you are not low-hanging fruit on the gullible tree and will be fighting them all the way. They will probably go as far as issuing a claim in the county court but you can be sure they will not see it through unless they want a spanking from the judge for bringing a case that has no hope of ever being successful. They are simply hoping that you will fold once they threaten litigation.