This thread has good info on this switch:
http://www.turbobuick.com/threads/no-power-to-the-purple-wire-on-starter.429380/page-2
Note that since you can turn the ignition key to crank the engine the linkage is likely adjusted OK. This is due to the P/N safety being a mechanical interlock on the column.
The switch at the base of the column operates the back up lights and the P/N input to the ECM. Over time the grease hardens and the switch stops working. It is easy to pull the switch, open it and clean it. Then a little new grease and good to go. The switch rackets left/right for adjustment.
The above link also has the wiring for the reverse lights to help trouble shoot what is wrong. To see if it is a switch adjustment issue do a key-on, engine-off. Then move the shifter from P to reverse and neutral. See if the backup lights comes on at any time. May need another person for this or a garage door after dark.
In my case there weren't any backup light bulbs installed...
RemoveBeforeFlight