Hi, welcome to JustAnswer!. This is Ed.
Operation and power flow on the park lamp circuit goes like this...
A multiplexed signal from the headlamp switch is received under the hood at the Front Control Module/ FCM, which is the silver box attached to the Power Distribution Center/ PDC, also sometimes called (collectively) the IPM. No matter.
The FCM takes the signal and switches the park lamp relay control circuit ON, which closes the relay contacts and energizes the fuse you were looking at. Nothing happens until the relay is energized and that's probably where things fell apart.
The IPM/ PDC/ fusebox unit has a vulnerability to water entry, which may decay the command signal between the FCM and the park lamp relay. This is especially true for the Rust Belt states, but we see this happen even in (mostly) dry Colorado. The exposed circuit is on a printed circuit board and since it was designed to carry very little current, it doesn't take much to rot this thin conductor and lose control of the park lamp relay.
True diagnostics of this sort involve the use of a scan tool, one capable of monitoring headlamp MUX signal, plus intended output control of the park lamp relay. There could be problems other than what I described and they wouldn't be possible to detect without this sort of "inside" information, but chances are heavily weighted toward a rotted control circuit.
Repair of the rotted circuit is possible, but not recommended. It involves complete disassembly of the IPM/ PDC unit and even then you might find the circuit to be beyond repair. The recommended repair is to replace the IPM/ PDC unit and transfer the FCM over to the new one. No programming is needed and the unit disconnects from the existing harness, so no soldering or loom replacement is needed. A water shield will be included with the replacement unit, showing the factory understands that corrosion is a likely cause of failure, but no recalls or extended warranties have been issued.
Let me know if you have any questions or problems and I'll be glad to help. A favorable rating is the only way I'm compensated for my time and that comes from the deposit you've already generously deposited.