There is a large multi-storey car park (Feethams) very close to the Dolphin Centre.  Drive up Beaumont Street and turn right through the automatic gates. There is now a ANPR (Automatic Number Plate Recognition) system in place which records your arrival time.  You pay on return to the car park.  The charge is now £1.20 per hour.

Alternatively you can find details of other nearby carparks here :-

Darlington BC - Car parks and charges