Dart 63 shift cable

The hesitation to go into gear could be indicative of a low fluid level. Get the car warm and level and check the fluid level. On our cars the torque converter fills itself in neutral and therefore should always be ready to pop into gear because we only have gears (R,D,2,1) and neutral. Park and neutral are the same.
Assuming your fluid level is OK, if the park lever or gear selector buttons are becoming difficult to move because it feels like the cables are hanging up, it is time to investigate. You can lube the entire park/gear selector assembly while it is still bolted under the dash with a quality silicone spray lube. It was recommended to use CRC extreme duty silicone. (Green can). All the metal slides and springs are probably terribly dirty and dusty after 54 years of use. Try to spray into the cables where the metal lines enter them under the dash as well.
Next check the route the lines take from the firewall to the trans and loook for obvious change interference. Should be a pretty smooth run from an oval shaped plug in the firewall to the gear selector(front of trans near kick down linkage) and the park cable will follow the same path as the speedometer cable and be routes through a metal bracket on the floor of the car before entering the parking sprag cover on the rear of the trans.
If All looks well, then see if the gear selector line is being cooked by the exhaust pipe. Usually that will result in a leak of trans fluid from the line, but maybe it just melted it enough to cause some binding. The buttons should push with some effort but should click into their respective gears and the trans should shift without delay. So something is wrong. Any other gear give you problems? Just reverse?