Park lamp fuse feeds the dash lights and the tail lights, so these are related issues.
The only thing that makes a fuse open is excessive current, which basically means a short to ground. The brown wire for the park lights comes out of that headlight switch, and it splices off and goes to every running light on the vehicle: front park, rear tail, side markers, clearance lights on the roof if you have them.... everything. There is a short on this circuit somewhere.
First thing to do is eliminate easy things. If you have a trailer wiring harness adapter (it sandwiches into the truck harness to give you lights for a trailer), remove that thing, reconnect the truck wiring the way it should be, and see if the problem goes away. If not, you're going to be looking for a shorted brown wire. Start at the lights, front and rear, and just give the wiring a visual inspection. Look at the rear frame where the lamp harness runs along the frame channel and near the trailer hitch. These are all areas where that harness can move around and rub on things. It's less likely to be inside the truck cab, but you can't rule anything out. I always start looking at the easiest places to access.
This is going to be a tough diagnosis, you simply have to put your eyes and hands on the wiring and find the shorted brown wire. The dash lights are going out because this brown wire is getting shorted somewhere, even if it's not blowing the fuse at that point, the amperage is being sucked out of the circuit wherever this short is. I would suspect a component over a shorted wire, due to how it's acting, but you can't rule out anything at this point, all we know is that the brown wire for the park lamps is short circuited somewhere.