Here is a quick start to get the gears turning...
ParkingLot is a class.
ParkingSpace is a class.
ParkingSpace has an Entrance.
Entrance has a location or more specifically, distance from Entrance.
ParkingLotSign is a class.
ParkingLot has a ParkingLotSign.
ParkingLot has a finite number of ParkingSpaces.
HandicappedParkingSpace is a subclass of ParkingSpace.
RegularParkingSpace is a subclass of ParkingSpace.
CompactParkingSpace is a subclass of ParkingSpace.
ParkingLot keeps array of ParkingSpaces, and a separate array of vacant ParkingSpaces in order of distance from its Entrance.
ParkingLotSign can be told to display "full", or "empty", or "blank/normal/partially occupied" by calling .Full(), .Empty() or .Normal()
Parker is a class.
Parker can Park().
Parker can Unpark().
Valet is a subclass of Parker that can call ParkingLot.FindVacantSpaceNearestEntrance(), which returns a ParkingSpace.
Parker has a ParkingSpace.
Parker can call ParkingSpace.Take() and ParkingSpace.Vacate().
Parker calls Entrance.Entering() and Entrance.Exiting() and ParkingSpace notifies ParkingLot when it is taken or vacated so that ParkingLot can determine if it is full or not. If it is newly full or newly empty or newly not full or empty, it should change the ParkingLotSign.Full() or ParkingLotSign.Empty() or ParkingLotSign.Normal().
HandicappedParker could be a subclass of Parker and CompactParker a subclass of Parker and RegularParker a subclass of Parker. (might be overkill, actually.)
In this solution, it is possible that Parker should be renamed to be Car.