Well, If you go down the road that Anne of Cleves House is on, and turn down one of the side streets, the parking is free in my experience - though admittedly I have not done it for a couple of years.
If you have people in the car who want a drink, park in Lewes (plenty of free parking near the station). There are loads of pubs to choose from, and the train takes about ten minutes to get to The Amex.