Man, that's a good looking '62 convertible! We want to see more of that car.
If you have disconnected the rod from the trans, and the trans goes through all the positions and goes into park, is the lever loose on the shaft going into the trans? That could account for a lot of play or not making it to park. If that's tight and the trans operates as it is supposed to from the trans, then either the rod length is wrong, the rod ends are worn out or gone, the lower level is hitting something as suggested or the problem is in the column. Check the gear selecting lever in the car without the rod attached to the lever on the lower column. Could be excessive wear or play with the tube in the column and the lever at the lower part of the column.