2006 GMC Savana parking light wont shut off

I can talk to you about the basics. The voltage to the park lamp relay is hot all the time. So, you need to check ground for the coil side of park lamp relay, that is a white wire. With headlamp and key off, You could use a digital multimeter, one lead connected to voltage, the other lead, touch the terminal for the white wire, if reading is battery voltage, the white wire is grounded for some reason, either through the bcm or the white wire is shorted to ground.

Also, you need to check the park lamp lamp turn on signal, goes from headlamp switch to bcm, brown wire with white tracer. At least for the park lamps, the headlamp switch is part of the ground circuit. You can take that wire loose from bcm and use the dmm the same way, if battery voltage, either that wire is shorted to ground or the headlamp switch has a problem? There are many ways to check the circuits? Others may have a better idea?

I can't rule out a bcm issue? Can you check for applicable trouble codes?

I just wanted to add, sometimes if you unplug a module, it can have unintended consequences, But, if your going to do an ohms test, I don't how else to do it, except unplug at both ends. I just didn't want to cause more problems than you already have.