Think you`ll find that long term parking parking in Gib is a big no-no..Although they are building multi storey car parks near the the new air terminal but
AFAIK they are not yet completed. The only other alternative would be La Linea. There is an underground one called Focona which is where all the hire cars are collected from and that is close to the bus station, about 10 mins walk from the border. I`m led to believe that there is another car park opposite the Iberostar Hotel on the Paseo Maritimo which is cheap and has 24 hour security, which I would have thought an essential considering La Linea`s reputation. Otherwise you might want to consider a lift or a taxi or the bus....
Might want to PM Compton 3 Bravo who unlike me uses Gib airport a lot and lives in that area and will have up todate info..