Section 79a of The Road Traffic Act of 1985 (as amended 2001) states:
(1) The Department may by order apply any provisions of
this Act, or of Part IV (traffic signs) of the Road Traffic Regulation
Act 1985, specified in the order to any parking place so specified, with
such exceptions, adaptations and modifications as may be so specified.
(2) An order under subsection (1) applying to a parking
place other than one provided by the Department or another public
authority under Part III of the Road Traffic Regulation Act 1985 shall
not be made without the consent of the owner of the parking place; but
where such an order contains a statement that it is made with such
consent, that statement shall be presumed to be true until the contrary
is proved.
In other words, for the last 10 years, public car-parks (like those at a supermarket) are classed as part of the public highway and all the rules of the road apply.