You are correct the neutral safety switch in mounted on the column for both shifter positions. If the car had headers installed on it at some point, the backdrive linkage would interfere with that linkage. Usually it was just removed, and the neutral safety switch would just be left in park. So the car would start.
My question is do your back up lights work? If not I think the neutral safety switch linkage is disconnected. If they do, then I think the bypassed the switch in the wiring.
Good luck, let us know what you find.