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.

I tested the brown wire, and it does not have any power. I tested it without the key in the ignition. But then I tested with the key in the ignition and got the same result. The only ones that have power are the orange, green and red with and without the key in the ignition. Does the brown wire need to have power?

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.

So do you think the headlamp switch is faulty even though it is new?

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.

Okay, I unplugged the headlamp switch, and the running lights turned off. I checked the brown wire, and it does not have power. I plugged the switch back and the running light turned back on. So, I bought another switch even though the current switch is new and plugged it and nothing changed the running lights turned on.

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.

The switch has 11 wires.

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 disabled the DRL and that didn't turn off the orange parking lights. Here are the pictures of the wires.

Here are more pictures.

Here's the last one.

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!

Thank you. I will check that and let you know my findings.

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".

The lights that stay on are the back lights on the instrument cluster, the lights on the ac/ heater control module and the orange parking/turn signal lights. I already tried taking the fuse 14 and 9 but they still stay on. Yes, the truck has the stock light system.

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?

The dash lights don't get dim at all when using the dimmer switch.

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.