One could potentially advance an argument based on Jopson vs Homeguard, too, although I'd have to re-read that judgement. OP, is the flat they were delivering to yours? If so, what does your lease say about parking and use of communal spaces etc.?
Before we get onto any protracted debate about the merits of PCM's claim against the removals company... We can advise on the legal position, but any decisions regarding the maintenance of good relations with the removals company is a matter for you to decide.
To answer your initial question - yes, there are potential grounds to appeal the charge. However, it would be for the removals company to do so, as they are the ones being pursued. As ixxy correctly notes, PCM will almost certainly reject any appeal regardless of merit, and the same goes for an appeal to the IAS. The removals company would potentially need to be prepared to defend the matter in the county court.
As for whether you have any responsibility - that would depend on what your contract with the removals company says. It would need to include some sort of clause indicating that you were responsible for any private parking charges incurred whilst delivering to you. That said, even if such a clause existed, I'd argue that such a clause may be an unfair term, depending on its wording, as it would potentially make you liable for the driver's negligence in failing to read and abide by the terms of parking at the site.
