My '62 Imperial had the pushbutton transmission, but no park position. There was a foot-operated parking brake that, as you mentioned, operated on the transmission tailstock.
Wow, much later than I had imagined. Dad's '50 Lincoln has the GM automatic which also lacks park, it has the big 'T' shaped handle beneath the dash that you just yank up then twist to lock. I just chuckled to picture Ford buying transmissions from GM back then since they didn't have an automatic ready for prime time.
Back when I was doing the road rally thing, we used to drain the fluid, fill with stock fluid and run it for a few days, then drain, fill with a synthetic and go. I used Red Line, AMSoil was another popular choice later. Granted, they were standard transmission cars but we pounded the pss outta them and never had a transmission failure.
Count your blessings, Frank: you could be driving an Audi, which would involve similarly-priced repairs to one system or another almost monthly.