OP, you have been asked repeatedly to post your 'appeal' to NPM, their reply(ies) and your correspondence with JustPark.
So far we've seen none of the above. Nosy Parker researched what appear to be JP's terms, but we need to see your correspondence.
You are asking the impossible of us without hard, objective facts. OP narrative, summaries and conclusions are a recipe for disaster.
Anyway, basic contract issues:
You have to attempt to mitigate your losses. Therefore you need to exhaust the creditor's(NPM) claims against you. If their response(s) state that you've reached the end of their internal procedures then, with our assistance, you need to decide whether to have their claim arbitrated. As this would be for the full charge in each case you would have to assess whether taking this further would be a sensible attempt to mitigate your losses. If, as you say but we've yet to see, NPM's case is sound then you should pay at the discount.
Of course you can claim against JP, it's NOT a small claims case or indeed any type of case, it's simply you explaining to JP what's gone wrong and what you've done in an attempt to resolve matters. If this were to fail then you would issue a Letter of Claim and formalise matters prior to, potentially, commencing proceedings.
But we need facts in the form of correspondence, not your summaries.