Hello, thank you for the extra information. Vehicle Control Services Ltd are members of both schemes i.e. the International Parking Community (IPC) and the British Parking Association (BPA). This is not rare. With members of the IPC the appeals go through the IAS and with BPA members it is through POPLA.
Your particular case is proceeding through the IAS so we will focus on that. It is normally advised not to appeal to the IAS which is widely considered a kangaroo court, and almost certainly would be rejected, despite what you say to them and will give them new impetus to pursue you even stronger and they might be buoyed by an IAS 'win' under their belt to brandish in case they issued.
However, this should not deter you if you have. From the information you have provided, there is sufficient evidence to dispute the charges. The main issue you are facing now is, they either accept what you are telling them, or if they do not and you refuse to pay the fine, they will need to issue court proceedings. As mentioned in my previous response, this is the only way for them to get money from you and you should expect them to be unreasonable, even aggressive in their pursuit of the fine.
The 10-minute grace period would not apply to your situation as it is a private car park which carries its own enforcement. The grace period only applies to regulated on-street parking bays or in a car park run by the council, so I would not focus on this aspect.
The truth of the matter is, as I anticipate it to be is all your appeals will be rejected and your concerns will fall on deaf ears. It is not right, but this is how they operate. The options will effectively be, to either pay the fine to conclude matters, or fight it....subject to them issuing court proceedings. You may try arbitration as that is quicker and more swifter than court proceedings but you essentially require either that or court to determine matters as opposed to any appeals than go through the IAS.
I appreciate it is frustrating but as mentioned, if it got to court, especially given the evidence you have to show the fine was unwarranted then you have good prospects to defend the matter.
I hope this helps.