With the country house at Mirehouse and Dodd Wood itself sharing the same car park, it's pot luck whether you'll get a space here during the summer months. The good news is that a little further along the A591 (driving away from Keswick) there a couple of places where you can generally manage to get parked. Of course you're then left with the need to follow the footpath back to the main car park, but needs must I suppose.
There is a charge at the main car park, but the alternatives along the road are free. There's also a good tea room at Dodd Wood.