There are various parking options,

 http://www.cityofbath.co.uk/Traveling_to_Bath/body_parking.html

Now with the new South Gate Shopping Centre across the street, this is now an added parking option - 876 spaces and a 3 minute walk from The Henry.

 To get to SouthGate Bath, simply drive into Bath city centre, then head along St James Parade until the end of the road, where you'll find the entrance to the SouthGate Bath car park.  

http://www.southgatebath.com/media/294020/car%20park%20map%20web.pdf

CAR PARK TARIFF:

Up to 2hrs   £3.00 /  Up to 3hrs   £4.00 /  Up to 4hrs   £5.00 / Up to 5hrs   £7.00

Up to 8hrs   £9.50 / 8hrs +         £12.00

After 6.30pm - 9.30am next morning £2.00

The car park is open 24 hours a day and is monitored by CCTV.