Quire a few variables here — where the signs were, how prominent they were, exactly what they said about “patrons only”….I don’t think it would be hard to make a plausible defence.
If you want proper advice, the MoneySavingExpert private parking forum is excellent. I’d post the notice there, with your personal details and registration number removed, together with photographs of the signs if possible.
Bear in mind that the parking company itself wants you to pay, so I wouldn’t place too much faith in the first-stage appeal process. If they reject an appeal, that certainly doesn’t mean they are legally right.
If it remains unpaid, the usual next stage is a series of increasingly demanding letters, sometimes from debt collectors. Debt collectors themselves don’t have any special enforcement powers — they can’t send bailiffs round or seize anything without the matter first going to court and the parking company obtaining a judgment.
Eventually they may threaten or actually issue a county court claim. Even then, you can defend it, and a judge will look at the evidence rather than simply accepting what the parking company says. It would them well over a year to get a court date.
If it did ultimately go to court and you lost, you would of course need to pay the amount ordered by the court. For a normal private parking case the total is usually around £300 hundred pounds.
What many people find is the parking company bombard you with letters, threaten court, issue proceedings and then abandon the case. Not all the time, but often.