Peel the potatoes and cut them into ¾-inch dice. Immediately place them in a large saucepan and add enough water to cover by 1 inch.
Boil water over medium-high heat.
Add 1 tablespoon salt, reduce heat to medium, and simmer. Stir a few times, until potatoes are fork tender, about 8 minutes.
Drain potatoes and transfer back to the pot.
Add vinegar and use a rubber spatula to toss gently to combine. Let stand for 20 minutes, the potatoes will still be warm.
In a large bowl, gently stir together potatoes, celery, mayonnaise, relish, onion, parsley, mustard, ¾ teaspoon salt, ½ teaspoon pepper, onion powder and chopped eggs. Add more salt and pepper to taste.
Cover and refrigerate until chilled, about 1 hour before serving. Garnish with more parsley, celery, and onion if desired.
Potato salad can be made up to two days ahead. Store in an airtight container in the refrigerator.