The driver has parked there many times before CE were contracted and was unaware there had been any change until the RK received the PCN. [and apparently disregarded the private parking signs which have been in situ for some time]
The entrance to the car park is not what you've shown as coming off the main road, this is Greenwood Vale, the car park entrance is immediately off to the right of Greenwood.
It's a tight turn and I suppose it's possible to miss the 3-foot high sign Private Parking which has been on the entrance fence for some time but not the equally large Turkis Private Parking sign immediately facing the driver as they enter. The one at the entrance also has an associated 'new parking rules' sign.
Your photos omit the large sign facing drivers as they enter, perhaps this has now gone, albeit that GSV shows this and similar going back years.
IMO, with what an assessor could see by looking at GSV and what we know is there now combined with what appears to be a long-time habit of the driver disregarding these signs I would not put any money on 'inadequate signage' being a winner.
I'm a grandparent myself and might also consider it somewhat heroic, almost a duty, to help my grandchildren when the time comes(hitherto restricted just to children) and if you're underwriting the £40 discount what's there for her to lose? But I cannot see any prospect of success based upon what appear to be the facts:
Car park has carried private parking signs for years, albeit with no teeth;
There are now teeth in the form of a private operator;
There is a large sign on the actual entrance which in daylight could not be missed by a reasonably attentive driver driving slowly as one would have to in order to negotiate the turn;
As observed by others, there is a host of signs around the car park setting out Ts and Cs.
IMO, at present she doesn't hold a winning hand, only a wish that in some way there would be a contractual issue which would help her. But you have no evidence in support, just a hope.