A mechanism for moving vehicles within a vehicle parking structure comprises a lift platform which can be moved vertically and laterally to positions in register with one or more transfer rooms through which vehicles enter and leave the structure and a selected parking space of the structure. A vehicle transfer mechanism is carried on the platform and includes four self-powered tractors carrying lift arms that can be driven off the platform to engage the tires of a vehicle and to transfer that vehicle onto the platform for transportation between a transfer room and a parking space. The tractors, when deployed, pass outside the vehicle tires with the lift arms folded and can then be operated to extend the arms in a way that engages the tires and lifts the vehicle.