Place the potatoes in a large pot, cover with cold salted water, and bring to a boil. Cook until fork-tender, about 12 to 15 minutes.
While the potatoes cook, hard-boil the eggs, then cool, peel, and chop them.
Drain the potatoes and let them cool slightly so they stay tender but do not fall apart.
In a large bowl, whisk together the mayonnaise, mustard, vinegar, sugar, salt, pepper, and paprika.
Add the potatoes, eggs, celery, onion, pickles, and parsley. Fold gently until evenly coated.
Chill for at least 30 minutes before serving for the best flavor.