At the risk of sounding like a jerk... I want you to try adjusting the dimmer switch first. I have had several people with problems very similar to yours where the end result was the dimmer was adjusted all the way down.
Now, if you adjusted the dimmer up and it still does not work I can step you through testing the lighting system if you have a meter or test light.
Ok, for starters check both legs of the INST LPS fuse located in the passenger side instrument panel fuse block. You should have battery voltage on both legs with the parking lights turned on.
Let me know the results of that test so we can continue on.
I don't have an INST LPS Fuse on the Passenger side instrument panel fuse block.
I do have a FRT PARK LP Fuse (Front Parking Lights, Instrumentation Lighting)
I cheked this fuse with the Parking Lights on and both sides have 12V to Ground.
Ok, I believe you are in the passenger side "underhood" box... not the "instrument panel" box. Under the hood is where you will find the PARK LPS fuse and relay.
On the passenger side of the instrument panel it should look like this:
I am in the Passenger Side instrument panel Fuse Box. Next to the glove box
inside the Passenger Side Door. It doesn't look anything like the image you sent.
It looks like this
Ok, sorry for the confusion. I just completely logged out of my GM schematics and started over from the beginning.
Passenger side - Check for power on both sides of the REAR PRK LP fuse.
Driver side - Check for power on both sides of the STR COL fuse.
Make sure the parking lights are turned on at the switch when doing this testing. Also, please verify if all exterior lights are working.
Rear PRK LP Fuse has 12V on both sides to Ground
STR COL I get nothing on both sides of the fuse
All exterior lights are working
Procedure for accessing the connector is shown below.
Ok, at the headlight switch test the Brown/White (terminal S) wire for battery voltage when the lights are turned on.
Then check for variable voltage on Gray (terminal J) wire with lights turned on and moving dimmer switch.
If you have power on Brown/White but not on Gray then the switch is bad.
If you have no power on Brown/White then the wire is bad from the fuse to the switch.
Connector Part Information
Headlamp Switch Signal
Park Lamp Switch Signal
Instrument Panel Lamp Supply Voltage - 1
Courtesy Lamp Switch Signal
Park Lamp Switch On Signal