Hello and thanks for using Just Answer.
My name is ***** ***** am happy to assist you with your enquiry.
Please be rest assured that you are not liable to pay any charge and certainly don't send them the £100. Although these private Companies threaten debt collection and Court action, they are never successful in any such proceedings provided the other party defends matters and nor are they legally entitled to demand any payment from you even if you have contravened the parking regulations.
From what you say, you have not even contravened the parking regulations in the car park, and you therefore need to write to them to appeal the fine, setting out the nature of your visits and evidence that you left and returned at the times stipulated by you.
I attach a link to the Money Saving Expert website, which has plenty of information on how to appeal.
I hope this assists and sets out the basic legal position.
In the meantime, please do not even think of paying the fine. A similar thing happened to me a few months ago and I successfully appealed my fine. Unfortunately, these Companies rely on the public just to pay up as sadly not many people know the law surrounding private parking fines (not that you have even breached "their rules" anyway.