IIRC for the purposes of tax and insurance, and licensing, if the public have relatively unrestricted access to it, it's "public".
A car park is normally generally freely accessible to the public from the roads, and as such would probably be counted as public.
A driveway to a house is private etc.
If it was behind a gate or some such that separated it from being generally accesable it would be private, if it isn't it's probably classed as public.
IIRC there was a case not long ago where a private estate was opened up for the day for an event, and one of the owners sons was done for driving without licence/insurance on his fathers land - normally it wouldn't have been a problem as normally the land was "private" but because the public had free access to it at that time it was classed as "public" in the eyes of the law.
So at a guess I would say it would be classed as being "on the road" for the purposes of the law, and as such would probably report it as being abandoned (I don't know why, but i've got a minor aversion to people leaving untaxed cars in public places, and a slightly larger aversion to them driving them on the roads).