To Therac's point, airplanes are not valet parked like cars. General Aviation aircraft operate out of Fixed Base Operators (FBOs) based on an airfield. These operations provide overnight parking, hangering, maintenance, fuel and other services to aircrews while on the ground. Once parked and shut down, the aircraft is typically chocked and left with the parking brake off to facilitate towing with a tug or similar airport vehicle.
When there are high volumes of GA traffic to events like this, FBOs may use a section of the ramp belonging to them for transient parking of aircraft. Generally the ramp is sterile and secure (and if they have a large volume of VIP aircraft, FBOs also employ their own security forces as well as local airport police to protect aircraft under their care. Most VIPs travel incognito for safety reasons; very few people know about their arrival or departure. The FBO staff would be in contact with the flight crew and let them know of where their aircraft was parked on the field. If a flight crew needed an aircraft immediately or had proposed departure time, the aircraft would then be towed from parking to an area on the ramp for pre-flighting, fueling, catering and loading for departure.
Below is a video Discovery networks did about 10 years ago or so on Donald Trump's 757 aircraft, which features the additional difficulties and arrangements for operations from SheltAir FBO at New York's LaGuardia airport.