Peel and chop the potatoes.
Soak potatoes in water, then drain and rinse the potatoes.
Add 2 quarts water, 2 tablespoons kosher salt, 2 tablespoons sugar, and 2 tablespoons vinegar to a large bowl. Add the potatoes to a stockpot and pour the salt, sugar & vinegar water over the potatoes. Bring to a boil over high heat.
Reduce to simmer, and cook, stirring occasionally, until potatoes are tender, about 10-15 minutes.
Drain and rinse the potatoes. Then transfer them to a rimmed baking sheet. Spread into even layer, then sprinkle with 2 tablespoons vinegar. Allow them to cool to room temperature, about 30 minutes.
Combine remaining vinegar, mayonnaise, mustard, green onions, relish, celery, dill, and parsley (if using), in a large bowl. Stir with a rubber spatula to combine.
Fold in the potatoes and chopped hard-boiled eggs (if using). Season to taste with salt and pepper.
Cover the rest in the fridge for at least 1 hour and up to 3 days before serving.