Ok, the part in bold implies to me that somewhere in the the truck you are losing connection to the trailer plug for your brake light circuit when the headlight switch is moved to the parking light setting.

I would start by looking at the headlight switch if I was troubleshooting this.

If it's cheap/easy enough maybe just replace it and see if that fixes your issue. Otherwise unplug it and start testing continuity across the various circuits as you move the switch through it's different settings.

Good luck on this!


<edit> LOL, didn't look at how old the OP was... I'll leave this here for the benefit of 6Mile. Maybe it will help you.