This might be a bit tricky in August, but the rest of the year you should find parking on normal streets and around:)
So here are two other choices:
Aparcamiento municipal de Tarifa.
Located next to the old town, location HERE: https://goo.gl/maps/f2Du4G84NRmVLh369
Prices and opening hours differ during high and low season. Usually closed from 8h to 1h in the morning. Until 4h the weekends.
Up to 1hour, 1€
8-12 hrs: 7.25€
Parking San Sebastian (covered and guarded parking)
Located in town centre, location HERE: https://goo.gl/maps/HNfkxbaqrJ3K3HGZ6
See all detailed information in their website, possibility to reserve: https://www.parkingsansebastian.es
Chances are that you should be able to find parking around where you are staying. The Old town though has some days and times that allow vehicles now.
So check the signs at the entrance.
Both the parking above are ok to enter all year :)