Unfortunately, this is FUBAR. You say that the driver has not been identified, yet here is clearly states that you identified as the driver:

Then you tell us that you agreed to pay two of the Parking Charge Notices (PCNs), yes they were indeed PCNs, in fact the two that were probably the easiest to defend, which is an admittance e of liability.
The one PCN where they have the best evidence of a breach of contract, a stay longer than the minimum consideration time, and therefore evidence of a contractual agreement, whether the driver read the signs or not, you are challenging.
I don't know what "gesture" you were trying to make by paying two of the PCNs except to mark you as a "mug" for further fleecing. You have gone about dealing with this in completely the wrong way and are now mired in potential litigation.
The two paid PCNs are completely irrelevant now except that the claimant could refer to them as evidence of the defendant already admitting liability on two pervious occasions for the same contravention at the same location.
Your only saving grace in the matter is that it is being handled by the incompetents at Moorside Legal, which, when they issue the inevitable claim, will screw up the Particulars of Claim (PoC) enough that you can hope for a defence that should win it on a technicality and the claim has a good chance of being struck out.
The one important piece of correspondence that matters right now, and we have not seen it yet, is the Letter of Claim (LoC). Sow that to us and suitable response can be provided, even though it will not stop them issuing the claim, you will be able to report Moorside Legal to the Solicitors Regulation Authority, should you be so inclined to do,