In a supermarket car park, are drivers obliged to comply with zebra crossings? (UK laws please)?

Okay, so it is a zebra crossing, with no belisha beacons, no zig zag white lines leading up to it etc, and it is on private land in a supermarket car park. Who has right of way? The pedestrian or the cars? Some people have told me that if there is no belisha beacons, it is a recommended place for pedestrians to cross, but there is no right of way for pedestrians. Is that true?