A carriage for handling a vehicle in an automatic parking system includes a frame axially movable along a first longitudinal axis, rotating clamps, each including a pair of counter-rotating arms rotating with respect to a vertical axis orthogonal to both the longitudinal axis and the transverse axis. The carriage includes a transverse-centering unit with transverse pushers, transversely extensible, moving along an axis parallel to the transverse axis and independent of the rotating clamps. In a first retracted position they do not interact with the vehicle wheels. When extended, they come into contact with at least one vehicle wheel. The extensible transverse pushers are on left and right sides of the carriage, and simultaneously move in a symmetrical extension, being operated by a respective motor positioned laterally of the longitudinal axis of the carriage. The extensible transverse pushers include a pantograph for longitudinally aligning the vehicle along the longitudinal axis.