I turn it off and take the key out the parking lights and instrument cluster lights stay on?

Can you remove the headlight switch and test and unplug it to see if you have power to the brown wire, please?

This guide will help:

https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester

Also, I have included the wiring diagrams so you can see how the running light system works. Check out the images (below). Please let us know what happens.

The brown wire is what supplies power to the running lights so if it doesn't have power the lights should be off. The only thing that supplies power to the brown wire is the headlamp switch. Please check the running light wiring diagrams above so you can see what I am talking about.

If you have power at the brown wire there are only two ways that can happen, either you have a bulb short that is feeding power into the brown wire or the switch is doing it, that is why I wanted you to unplug the switch to see if it still occurs. Please let me know.

Can I ask how many wires the switch has? My info says 4. Here are the headlamp wiring diagrams as well, the system is pretty straightforward. As you can see the headlight switch supplies the power for the running light system, how is the wiring, has it been tampered with? Check the images below.

Okay, can you please upload a picture of the switch and wiring? Also, can you unplug the DRL module to see if the lights go out? Here is the location of the module in the images below. Let me know.

I will ask another expert to help out.

It's possible the harness has rubbed through on a piece of metal maybe behind the dash, finding a new path to ground and by-passing the switch. Visually inspect the harness around the switch area and follow all the wires you can. If you don't see anything obvious, I would unplug the switch and back-probe the connector using a DVOM and some t-pins to check for extra resistance on each wire. Make sure to disconnect the circuit completely to avoid inducing voltage to any modules. If not, you can fry a module. I bet there is a bare wire somewhere. Just my 2 cents. Hope it helps!

Reading through this which dash lights are on? The ones that backlight the dash? Those are connected to the radio lights and others. They get power through fuse 14 illum.
You say the parking/running are also on? Those are fed through fuse 9 and that fuse also powers some of the illumination on the switches like power windows and ash tray lamp. If you pull either of those, do you see either of the lights, you have a problem with turning off?
If one does it will isolate some of the system.
This truck has all stock style light bulbs in it, correct? No LED "upgrades".

Pull fuse 14 only. Do the dash back lights do out? Asking because fuse 9 powers fuse 14 and the two circuits split off to feed different lights with no other connections, if pulling 14 does nothing and you still have the dash lighting and markers on as well then you have a short that is feeding power and will likely need to access the main harness behind the dash to find it. The brown wires Ken mentioned feed the parker lights, but the dash lights are on a different circuit and use gray wiring. Does dimming the dash lights do anything?

Okay, if the dimmer does nothing and pulling fuse 14 changes nothing then you have a short to power. You say that when you unplugged the switch the parking lights went out, did the dash lights also go out? Try removing the 50-amp maxi fuse 8 from the under-hood box, do the lights go out? If yes put it back in, then pull fuse 15, any change? Pull fuse 3 anything?
Trying to determine what circuit is feeding power to the wrong places.