Cover potatoes with cold salted water by 1 inch. Boil, then simmer until tender, 10–12 minutes.
Drain potatoes and spread on a sheet pan to steam-dry until barely warm.
Hard-boil eggs: cover with water, bring to a boil, cover, rest 10 minutes, then ice-bath, peel, and chop.
Whisk mayonnaise, mustard, vinegar, sugar, celery seed, salt, and pepper in a large bowl.
Fold in potatoes, chopped eggs, celery, red onion, and relish until coated. Adjust salt to taste.
Chill covered at least 30 minutes. Sprinkle with parsley and paprika before serving.