The blue badge scheme is not valid in car parks, it says so in the booklet that comes with it. The blue badge scheme is only valid on public highways.
The Equality Act is what is valid in car parks and private land. Anyone who qualifies as disabled under the equality act can have reasonable adjustments.
In fact a car park stating only blue badge holders in disabled spaces, or blue badge only spaces is disability discrimination. This has been tested in court many times.
A person who is diagnosed with cancer is considered disabled under the Equality Act from the moment they have been diagnosed. But quite a lot of people with cancer don’t get blue badges. With cancer treatment it often makes you exhausted and you need the spaces closer to the shops.
There are various reasons a disabled person needs a disabled space on private land and even if you have a blue badge you don’t need to display it.
If you get a PCN for not displaying a blue badge on private land or car parks then sue them for disability discrimination.