forums.moneysavingexpert.com/forumdisplay.php?f=163&order=desc&page=2
This is the best place you will find advice. I post there regularly as "Loadsofchildren123". Don't look at the MSE site, which contains the wrong advice, but at the private parking tickets forum (that's the link I've put above).
First port of call is to try to get the surgery to deal with it. You were a genuine visitor and complied with the t&cs in every respect apart from this genuine and understandable mistake.
The surgery may just tell you their hands are tied. Don't take no for an answer, make a formal complaint via the practice manager and speak to your GP.
CEL's MO is to chase and chase you, to issue proceedings, and then to withdraw once you put up a good fight. This is different to other private parking companies who fight all the way, even when they don't have a case, because the small number of cases that fight are just loss leaders. At least CEL bows out when it can see it's going to court. So this is good news for you.
But this of course means that you have to educate yourself about how to approach the matter at every stage.
If the PPC does not know who the driver is, be careful not to identify who the driver was and talk in the third person about "the keeper" and "the driver". If CEL cannot identify the driver (which they can't) then they have to jump through various hoops to make you liable as keeper. If they fall at this hurdle, it's job done if they can't prove who the driver was. All of this makes it a bit tricky when you are dealing with the surgery, but just keep referring to these parties in the third person.
You have a time limit to appeal directly to CEL (again appeal as keeper and describe the driver in the third person). CEL will reject your appeal, but they will give you a POPLA code (this is a supposedly independent appeal). You then do a 2nd appeal to POPLA. You may not win at POPLA if you entered the wrong reg. POPLA apply a reverse burden of proof where it's up to you to prove a defence, rather than the parking company to prove its case. Again, don't be disheartened by a POPLA loss, you are just jumping through all the hoops. The MSE forum guides you through the process and what to say in an appeal.
If you don't win at POPLA then you have to wait for a claim then defend it. The claim is based on you having entered into a contract by parking in the carpark, the terms of which are set out on the signs, and breaching those terms by entering the wrong reg/failing to display a valid ticket. So the signs have to be prominently displayed, with clear wording capable of forming a contract.
Did you pay? If so, the wrong reg may be treated by a judge as a human error that doesn't amount to a fundamental breach of the contract (you complied in every other way), and you argue that the breach was "de minimis" and the case should be thrown out - so this needs to be pointed out to CEL as soon as you can.
Go onto the MSE forum and start a thread, you'll get lots of good advice and help. But you will be asked to read the step by step guides in the "Newbies" thread, and to educate yourself. The more willing you show in that regard, the more help you'll get.