Brake light fuse

That is either going to be a wiring issue or an issue with the brake light switch. The switch is located down below the dash, where the top of the brake pedal arm is. Remove the switch and check it for continuity between the lead and ground wires using a Multi-meter or an Ohmmeter. If there is any resistance less than infinity, then you have identified an electrical short. Be sure and check it while pushing the plunger all the way in and out several times.

Remove the stop lamps fuse, place an ammeter into the empty socket, press the brake pedal down and read the ammeter. If it is over what the fuse is rated at, remove all the stop lamp bulbs one at a time checking the ammeter after removing each bulb. If and when after removing a bulb if the ammeter drops safely below what the fuse is rated at, replace that bulb and check that socket and/or wiring for a short if the new bulb does not fix it.
It after no bulbs in, the ammeter still reads high disconnect each item here for fuse #13.

Ditto what PRP said. The problem is likely somewhere in the wiring, or possibly in that relay.

The other running lights should also stay on because they are on the same circuit. Are they?
I do not see any relays in the wiring diagram for the parking (running) lights.

Perhaps you have a short circuit in between the running lights and whatever that relay powers. Try to narrow down which relay it is, either put your finger tips on them and feel which one clicks when someone else turns the headlight switch on and off or remove them one at a time and listen if you are by yourself.

Was you sure to turn the key to run?

Those 2 fuses are just for the trailer lights.
I see them listed as fuses 6 and 7 in my book.
Do you have voltage coming out of the stop lamp switch when the pedal is pressed down? If so, check again leaving the multifunction switch.