Add 2 quarts water to a large saucepan. Add potatoes, 2 tablespoons kosher salt, 2 tablespoons sugar, and 2 tablespoons vinegar. Bring to a boil over high heat. Reduce to simmer, and cook, stirring occasionally, until potatoes are tender, about 10 minutes. Drain potatoes and transfer to rimmed baking sheet. Spread into even layer, then sprinkle with 2 tablespoons vinegar. Allow to cool to room temperature, about 30 minutes.
In a large bowl, combine remaining 2 tablespoons sugar, remaining 2 tablespoons vinegar, celery, onion, scallion (if using), parsley (if using), pickles, mustard, and mayonnaise. Stir with rubber spatula to combine. Fold in potatoes. Season to taste with salt and pepper. Cover and rest in fridge for at least 1 hour and up to 3 days before serving.
