Thats a shame as you would have walked away with nothing.
This is private land fine. They run against the driver not the registered keeper. They are issued by private land companies who have no more right to punish you for the manner of your parking than your neighbours do. Only the police or the local Council can do that. These are issued on the basis that you have accepted a contract by parking there and breached it by saying too long. Alternatively they can argue trespass.
In either case though, they have to identify the driver. This is not a crime but a civil dispute so they cannot use S172 RTA to force from you the name of the driver. They have no power to make you so you really should have refused that information and then they cannot find the right person to sue.
Having admitted it though, it is likely they will seize upon that. They used not to sue those who had admitted being the driver but so many people have worked out that these fines are unenforceable that they are becoming more aggressive now. Probably there's about a 50% chance they will sue.
Its probably still worth fighting just by simply putting them to prove about adequacy of signs and their wording. Half the time they are void.
Also, you could argue that the sum they are charging doesn't represent their true loss although I have to say I've never won on that point yet.
If you tell me what the signs said I might be able to give you a stronger indication.
Hope this helps. Please remember to rate my service OK or above.