Got it put together runs and drives but no Park.
Found this bit of information from what appears to be a reliable source related to a TDI
The reason parking mechanism isn't working because there is small spring that is pushed against small weak metal tab in parking paw assembly, it tends to bent out enough to weakening spring tension on paw then eventually will break off and cause parking paw to free float without locking differential. This usually happens from people putting transmission into P position without engaging Parking brakes on first before releasing brake pedal after putting car into Park, and cause excessive strain on parking paw at differential, when car rolls down with its own weight.
Bad news is there are no replacement parts you can get to replace that parking paw mechanism or broken small tab that holds spring in place. So only options are either replace whole transmission or try find correct metal material and fabricate tab yourself.
Regarding the codes you are getting, those are 2 sensors that are tied together at long stem of Mechatronic unit, they read output shafts speed that is connected to differential gear.
As far as why car doesn't move (freewheeling) when engaged into gear, I'm not sure what would cause that without having physically be there. I don't think it has broken worn out gears that cause slipping on output shaft, or destroyed differential. My guess would be broken tap from parking mechanism fall into and possibly damaged one of sender wheels for shafts speed or sensors itself on mechatronic unit.
Here is what I would do if I were you, talk to customer and ask him if he was parking car on hill, and if he was using parking brake each time he put car into park or was he would shift car into park and let car roll bit till park paw locks differential? Then using VCDS try clear the codes if they don't clear that's ok, next is let DSG fluid warm up to over 30C then on ross-tech.com find DSG adaptation procedure and perform that step by step and then see if transmission would engage into gear and spin wheels now. If not, then remove transmission and check DMF and remove mechatronics unit and inspect for any debris or broken parts. Then from there can decide if you want to tear down completely (split open case) or just get another transmission. BTW you can pretty much get any year TDI DSG transmission as long as you use 06 DSG mechatronic unit (if original mechatronics is good then I would reuse it if you find great deal on newer TDI DSG), to my knowledge you can even use gasser DSG as long as you put 06 TDI mechatronics (in next couple weeks I will be able to confirm this for sure).
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