Where can I park in Thame near the start of the Phoenix trail
Parking depends on the time and day you go, and at what point you intend to join the trail. At the end nearest the Lord Bill's Upper School in Thame, you could try parking in the school's top car park near the main road on a non-school day, or out of school hours.
Not sure about the Princes Risborough end parking (try looking at GoogleMaps), although you could join the Phoenix Trail in Thame near the old railway crossing near BOC on the 'eastern Bypass'. Near that point is an essentially a private car park for the old Leyland DAF offices, but you might sneak into a space there. I don't think that they use all those spaces for workers' cars as the facility has downsized a lot in the past years.
Get quick answers from The Phoenix Trail staff and past visitors.