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, taking a ticket as you do so.  The ticket allows you back into the car park and is required to pay on return to the car park.  The charge is now £1 per hour.

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

Darlington BC - Car parks and charges