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We still don't have answers to the following:

The 14-minute gym session, and
The elapsed times of the other PCNs.

Presumably if you are a teacher and are going abroad then you have a job, employer and place of residence. You might care to share this with the creditor now and say that you'll update them in the near future with the effective date on which all correspondence must be sent to this address. There is no reason for your parents' address (and them potentially) to stay in the firing line.

Further, yes I would much rather have this sorted before I leave the country especially due to the assets I am leaving.

You can seek to use dodgy timings to cast doubt on the reliability of the parking company's evidence. One potential pitfall to navigate is how to do this in a way that helps - an argument that comes across as "I was actually parked where the parking company say I shouldn't have been for longer than they claim" might not help your case. But a case that uses the fact that their evidence doesn't seem to correspond accurately with the times the car was there, alongside the other arguments around signage and lack of contract (forbidding signage) might be more compelling.

I note you mention of being keen for the matter to be resolved before you move abroad - unless you have any luck with the IAS (which is rare), I'd be very surprised if the matter is resolved before then. Whether the company do issue a claim remains to be seen - I'm not sure how litigious they normally are, although 10 tickets can tempt them.

H C Andersen, the OP has now addressed the issue of the timings on the PCNs four separate times:

The driver arrives around 6:10am and leaves around 8am each day.

As mentioned in other comments, the time gap seems to be around 10-14 minutes in terms of picture taken. To me, it reads as if perhaps they give an 'allowance' of some sort. The contravention time stamp is marked as the latter time. I, of course, stay at the gym longer than this.

I was not in the gym for 14-minutes.

But yes, the camera seems to be erroneous.

Unless they keep a written log of exactly when they enter and leave the gym each day, it's unlikely you're going to get any more specific timings than those the OP has already offered.

There is clearly an issue with the 'from' and 'to' times. From the photos, and the information provided thus far, it seems that the parking operator is using CCTV, rather than any sort of automated ANPR system. What I would suspect is happening (although this is of course my reckoning rather than necessarily fact), is that there is an operator manually checking the footage - they capture an image when they spot a vehicle parking up that shouldn't be there, then snap another once it's been there longer than the consideration period they offer (say, 10 minutes), essentially doing what a warden would do before slapping a ticket on the windscreen, but here doing it all remotely.

The person liable is the driver.

Uniquely, statute in the form of [Schedule 4 to] the Protection of Freedoms Act 2012 provides that 'the creditor will (if all the applicable conditions under this Schedule are met) have the right to recover from the keeper so much of that amount [the parking charge] as remains unpaid'.

Nothing to do with the circumstances of the events, this is purely procedural.

The person liable is the driver.

Did the appeals name the driver?

Frankly, from the points we've discussed so far, the one that seems to have the most legs in my view is one around the prohibitive nature of the signage, meaning that no contract was formed.