Richmond Park will be beautiful at that time in the morning, and not busy.
Parking is free at all the car parks (for the time being at least – the Royal Parks are considering introducing charges); Pembroke Lodge is the closest car park to Richmond Gate (Star and Garter) – enter the park and turn right, the car park is about 300 yard down the road on the right, and they do good tea and cakes.
Officially, you’re supposed to stick to the gravel paths around the park, but you can cut off and go where you like. There’s plenty of twisty tracks through woods in the middle of the park, and they’ll probably be plenty of deer about.
From Robin Hood Gate (SE corner), you can cross the A3 and head in to Wimbledon Common for a bit more exploring.
For local tracks in London (well, local to me), it’s really not bad at all.
Nick