Peel (or not) and cut the potatoes into bite-size cubes. Place the potatoes, 2 tablespoons kosher salt, 2 tablespoons cider vinegar and 2 tablespoons sugar in a large saucepan and cover with cold water by 2-inches. Stir well. Bring the water to a boil, reduce heat to a simmer, and cook until potatoes are tender, about 15-20 minutes. Drain and transfer potatoes to a large mixing bowl. Add 2 tablespoons vinegar and 1 teaspoon kosher salt, stir vigorously to rough up the potato edges slightly and cool to room temperature. While potatoes cool, prepare the other ingredients.
Use a fork to mash 3 of the hard-cooked eggs and rough-chop the other 3 (set the 3 chopped eggs aside to add later).
In a different large mixing bowl, add the remaining 2 tablespoons sugar, 2 tablespoons vinegar, mashed eggs, chopped celery, & onions, mayonnaise, relish (if using), mustard, black pepper, seasoned salt, paprika and parsley.
Gently fold in the cooled potatoes and remaining rough-chopped eggs.
If more salt and pepper are needed, add it now and stir gently. Keep in an airtight container in the fridge until ready to serve (1-24 hours).
When ready to serve, top with the sliced green onions for garnish.