I have used these people www.cophallparkinggatwick.co.uk/ several times and they have been totally faultless. In fact once there were travel problems round the M25 and they went out of their way to work out a route home for me avoiding the worst.
look on the parking website to see what deals they have. we have found valet parking for less than parking yourself in the past so always worth searching for this. a bus around rome is fab and we took the train out to the ruins.