From the factory wiring diagram:

If you don't have a copy of the wiring diagram, download it from this site. You can find the Download section after you click on Resources at the top of the page.

Update: the parking lights harness' 15A blade fuse was partially split. I glanced at it at the beginning, and confirmed it yesterday that it was indeed broken. I replaced it and the running lights are working again.

Still no side marker lights though. Plan is to replace the side marker housings, and see which wires are faulty when the 15A fuses "splits" again. 

 

@SteveJ do you have the parking light harness too? And all replacement LEDs (and electronic flashers) you listed have no issues with it?

 

Thank you so much for your help!

I don't have that relay harness, just the stock harness for that circuit. The only issues I experienced with some LEDs is that I had some type 1157 red LEDs fail so that they lit both "elements" at once. I chose a different brand that's in the list and experienced no problems since then. I first started using LEDs over 13 years ago and have no regrets.

You may find corrosion in some or all of your side markers. I think that is part of the issue with the circuit overall.