Ok so I replaced the park neutral position relay. Background story:
1997 A4 auto FWD. Prior transmission died a horrible death so a known good transmission was swapped in.

After replacing the transmission, the car would not start unless jumped at the starter. I had a helper turn the key while I checked for voltage at the starter solenoid wire. No power, which matched the symptom of not starting with the key turn. I had them try three times, on the third time smoke from the park neutral relay. I had to pull it fast. Relay was toast and socket had a little damage.

Fast forward to now. I was suggested to replace the transmissions position sensor (aka multi function switch). I did that today, with a known good one and also put a new park neutral position relay in. No luck when turning the key, and I pulled the relay to find one pin ridiculously hot. On the relay one leg is labeled 87, that's the offender.

Now my obvious question is where to look next? Prior to the transmission change, it started fine from the key. I R&Rd the starter as needed to change the transmission. Could this be a solenoid gone bad? A short in the solenoid circuit or something else? I tried to determine what leg 87 of the park neutral position relay goes to without success. I am pretty close to giving up after all the trouble with this car. Any insight greatly appreciated.