This is one of the most laughably inadequate NtKs I've ever seen. Weirdly they don't seem to claim in the NtK or even on their website to be a member of either the BPA or IPC (which I think means they shouldn't be able to get your data from the DVLA) but they are listed on the IPC website as an approved operator.
Regardless, the NtK is not PoFA compliant - so they cannot transfer liability to you as the keeper, and you have no obligation to name the driver.
Obviously it's also ludicrous to state an offence of 'No ticket displayed' if it's a no ticket car park.
I assume you've typed the PCN number and your VRM into the website to see if it will let you appeal? Would also be interested in whether there are any photos (and whether the windscreen ticket is shown).
Assuming you can't appeal through the site, I'd be inclined to appeal to their info@ or by post with proof of postage, for the sake of having done something if it ever went to court (using the standard wording repeated around the forum, add the no ticket point, ask for proof of NtD windscreen ticket), but rest assured you will never pay these people a penny.