The neutral safety switch has nothing to do with the shift interlock system.
Cannot shift out of the park position. The Center High Mounted Stop Light (CHMSL) is not working. The outer brake lights work if the ignition switch is on.
1. Check if fuse #24 in the TIPM (Totally Intergrated Power Module) is good.
2. Check that the Tan/Red wire at the brake switch has power at all times. Load test this circuit using a spare headlight.
3. Check that there is B+ on the White/Tan wire at the brake switch when the brakes are applied. If not, the switch is bad.
4. The instrument cluster sends out B+ on the Orange/White wire to the BTSI solenoid when the brakes are released. The cluster gets voltage on the White/Tan wire when the brakes are applied and stops sending voltage to the BTSI to release the shifter. The White/Tan wire also goes to the CHMSL.
The outer brake lights get power from the TIPM. The TIPM gets a bussed message from the cluster when the brake pedal is applied. If the cluster does not get power on the White/Tan wire but the Powertrain Control Module (PCM) and Controller Antilock Brakes (CAB) get a signal that the brakes are applied, one of the modules sends a bussed message to the TIPM as a failsafe. This failsafe only works when the key is turned on.