I am in the midst of a dispute with Parking eye.
I was at a conference in a hotel in Bournemouth and was told to where to park by a receptionist and that as I was attending the conference I would not have to pay. A notice on the Reception desk confirmed this. A month later I received a demand for payment of a fine from Parking eye. I told them (by letter and email) of the circumstances but this so called 'appeal' was rejected. I have refused to pay and regular letters threatening increasing fines, payment of their legal costs and court action have followed. For a while I responded to each one in writing but I now throw their letters in the bin. This was two years ago and whilst I haven't heard from them for a couple of months, I have no doubt their computer will throw out another threatening letter soon.
I am fully prepared to go to court and counter sue for harassment. I don't know why this has particularly got under my skin but it has and because I am perhaps young enough to not feel intimidated and I have adequate means to fight a court case, I will do so. However I suspect it will never happen because Parking Eye's tactic is to worry people into paying rather than risk court action.
If there is a clear cut case they might take it further and will no doubt win. However if there is any doubt (and there clearly is in your case) I do not believe they will. However it is profitable for them to send out continuous letters threatening action because many people will pay either out of fear or just to make the subject 'go away'.
So my advice from my own experience is to just ignore them and eventually they will give up. They are bullies operating on the very edge of the law but they will, like all bullies, eventually turn their attention to the easier targets (which is morally wrong again as the 'easier targets' are likely to be the elderly) and give up on those that stand up to them.
I would recommend filing the letters rather than throwing them away!