Sorry, but Parsons Branch is 1 way, and not going the way you want to go. it is pretty cool, you start in the valley and cross at least a couple ridge line before you come out in the middle of the Dragon.
Also check with the Rangers to make sure that it is open all the way. We went through there last year, bout this time, and there was a bunch of dead (still standing) trees. All you need is one big storm with wind and that road will be closed for a bit of time. I do remember something on the news about it being closed a couple months ago, so might wanna check to see if it reopend.