Positive outcome update:
Just an update on this and a thank you to those who contributed points, esp oz220 - whose detailed advice I took as a useful suggestion of an approach to take (basically little point trying to ask the parking contractor to further engage, go to the TOC that they are contracted to and appeal to their better nature - in this case Southern).
So I wrote to TSGN explaining the situation and the extenuating circumstances relating to my mum and my dad, as her carer, in the hope they would look at this again, given a car park ticket had been purchased all along and there was no intention to evade payment.
Although I have not as yet had an actual reply to that letter from TSGN**, the parking contractor subsequently contacted my father to say they had been asked to refund the penalty as a 'goodwill gesture'. It is of course reasonable to assume this was after Southern had asked them to do this.
I am therefore very grateful to Southern, and whoever on their staff took the trouble to read my letter, consider it and act on it in a positive manner.
I would like to say many thanks to forum members who posted in the thread offering suggestions, which helped me weigh up the best method of asking for the case to re-considered.
** EDIT: A few days after posting this I also received a helpful and sympathetic reply from TSGN, for which I was grateful, and that also allows me to the thank the staff involved directly.