Welcome to FTLA.
To help us provide the best advice, please read the following thread carefully and provide as much of the information it asks for as you are able to:
READ THIS FIRST - Private Parking Charges Forum guide by law I shouldn’t need to display a blue badge on private car parks
By which law? On private land, the owners of the land can impose whatever requirements they like.
They
do have to make "reasonable adjustments" for disabled people under the Equality Act, although whether a reasonable adjustment for someone with autism is allowing them to park closer to the gym than others might be a stretch.
The good news is that DCB Legal very often back down when a claim is defended. Show us the relevant documents as detailed in the link above and we can advise.