This just happened to me today as I was driving from Virginia back to Nebraska, and for most of the trip, the car was angry. I'll just let the video I posted tell the tale; first some information, this is on an automatic equipped car, and the sound doesn't occur when the transmission is in neutral. Based on some videos I've found of cars with faulty ring and pinions, this sounds like it's the case here.
A couple things, the part number for the ring and pinion is 087-409-143-J, which is NLA from most places, though I found a VW dealer that sells them for just shy of $1K
Do the R&Ps from the manual transmissions fit the automatics, or better yet, I hear these are Audi boxes... does anyone happen to know about these transmissions and where I could snatch a fitting R&P? I'll be taking the car to a transmission shop to get it diagnosed and taken care of, and if I can supply the parts, all the better.
https://www.youtube.com/watch?v=pm24bz-Y3Ow