There is usually no border queue for Gib on National day ... it's a bank holiday so there are no/very few commuters. No locals leaving, and not many Spanish arriving!
If you decide to cross the border in your hire car (usually OK with ANY hire company), you will be able to park opposite the airport for about 50p per hour, and take the bus into town which are free of charge on National Day!
If you want to park in La Linea I would recommend either opposite the bus station (2 euros per day - closes at 8pm) or opposite the Iberostar Hotel - same price - open 24 hours.
Don't forget to wear red and white!
And - enjoy - Gibraltarians could really show the UK a thing or two about being 'proud' to be British!!