This is the video of the 11+ year old system installed in Århus Denmark by Lödige Industries from Germany. Lödige Industries more recent solution is actually moving the whole pallet you park on, so you can also plug your car into it: https://youtu.be/-_ZxnDrfwGk?t=46
The system in London Gatwick UK is done by Stanley Robotics from France (which was acquired by HL Robotics from South Korea in 2024), but it could be similar.
Sorry yeah, I understood that they're not the same company or system, but I couldn't think of any way it could be done (aside from, as you say, parking already on it) so it was nice to see at least one solution.