Yes - there are 3rd party lots - usually off the airport grounds. We used one on our honeymoon - it was part of the package - we pull in they park our car (after taking everything out) put everything in a van and drive us to the airport door. They unload for you and you go in and fly away. When coming back home - you give them a call 24x7 when you land and they pick you up at the door do the opposite. Car is waiting for you and running.
Overall it was a good experience when we used the service (different company than OPs company). Nice thing is - if there is a problem with the car - no need to wait and flag someone down.
