You're refering to the Herman Goring Panzer Division. It wasn't Fallschirmjagers though, just an elite Luftwaffe Ground unit (unlike the other Lufty Ground divisions). The closest things the FJ got in terms of tanks were Stug3Gs in 1944.
As for PzGrn, they were in Heer, SS, and Luftwaffe. Any branch that had a panzer division, basically.
