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Is it a floor shift now? Probably was a column shifter before and they changed the steering column out for a floor shift one and never ran the wires to the floor shifter. May have been just too lazy to run new wires to the floor shift and just connected your column shift wires together. Does it start in reverse also?
Hopefully it isn't that and only an alignment issue with the neutral safety switch.
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.