first you need to install 180 degree diverters to keep the water off the fence. This is the chlorine and floride in the water doing this.
what will clean this are: product from the flood company sold in Sherwin william;s stores. Culprinol makes a fence and deck cleaner and if the fence is red cedar you will need to use a brightener after the cleaner. If the fence is NOT red cedar, the brightener is not needed.
You get a manual pump deck sprayer and spray the chemical on the areas and after about 15 minutes, then wash the fence down with fresh water thoroughly. Then you re stain the area.
When I say area, I would not only clean the affected area, but that entire panel as you will get a clean area and an old stained area. This will work as I have used it many times for just the same thing.
get those water diverters on your sprinkler heads.