Yes, there is a "park", but it's hidden. With the engine stopped, put the shifter in Reverse. The driveshaft locks up. Yes, there should be a neutral safety switch, probably somewhere on the steering column. The first Hydramatic to have a Park position on the shift quadrant was the "Controlled Coupling" model that came out in 1956, but it was only used in cars, not Chevy or GMC trucks. Have fun with that one.
BTW, the throttle pressure adjustment (linkage from the carburetor to the transmission) is VERY sensitive. A tiny change in the linkage length makes a huge difference in the shift strategy of the transmission. It's normal to get a 1-2 shift at very low speed- - - -5-10 MPH is about average. The 2-3 shift is the one that the throttle linkage adjustment has the most noticeable effect on. There should also be a 4-3 kickdown shift at full throttle.
Jerry
Is that “controlled coupling” tranny similar to those BOP Fluid Drive units? Where direct drive is all fluid and no gear?
