A public place is a place to which the public have access to on payment or otherwise, case law stated ages ago that a car park is a public place and so the Road Traffic Act applies.
The old one entrance at one side and exit at the other doesn't apply anymore, AFAIK.
Just because its not registered or conforming to construction and use regulations doesn't mean that it isn't classified as a mechanically propelled vehicle.
So, riding a mini-moto on a car park, or a car on a car park is the same for the purpose of the RTA.
And riding on a car park and a road is the same for the purpose of the RTA.
Edit - the link doesn't mention a car park
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