Might be worthwhile contacting the Mountain Rescue Team (Lomond) who are based in Drymen and seeing if they would let you park your van in their car park for the week. The least you can get is a NO
Failing that I've a few friends in Drymen who might let you park outside their house if you're really stuck.
My concern is how you would get back to Drymen after the walk as you might be best getting the train back and then would need to get a bus from Balloch. In this case it might be better parking at the National Park Headquarters in Balloch (free public car park) and then getting a bus to Drymen before the walk rather than having to get a bus after the walk when you're knackered. The bus stop is directly opposite the NP headquarters. This would probably save about an hour and a half on your return journey from Pitlochry.
Let me know if you need any more help and I should be able to sort something out..