My OH usually uses the Seven Springs car park, which is also good for the stepping stones and some nice canter tracks. It's best to avoid sunny Sunday/Bank Holiday afternoons at the busiest places for ease of parking, and lessening the risk of getting tangled up with dogs and children. Early mornings are usually good anywhere on the Chase. Most bike users are very good and call out if they come up fast from behind, but there's the odd few who don't, and they can approach very quietly indeed.
You can get very close to deer when riding. Some horses don't like them, others don't bother at all.
The Chase is really a multi-use area. Pick your time and area right, and you may come across the husky drivers! I am one of a few carriage drivers who get up there from time to time.
I think, strictly speaking, you are meant to have a permit to go on some parts of the Forestry Commission area. I have one, because I need a key to open the barriers, but I have seen people ride past them without the identifying hat band. There are, in any case, a number of bridleways that cross both County Council and Forestry Commission land.
It's a gorgeous area to ride in, so do come and enjoy