The status of the car park is really of no consequence here. The Road Traffic Act (Section 143) requires a driver to have Third Party cover when driving on "a road or other public place." The surgery car park will almost certainly meet this description. But even if it did not, a driver at fault is still liable to pay for damage caused even if the incident took place on a private estate to which the public has no access. The only difference is he does not need insurance cover.