In a 6- to 8-quart pan, combine potatoes and 4 quarts water. Cover and bring to a boil over high heat. Reduce heat and simmer until potatoes are tender when pierced, 25 to 30 minutes. Drain and let stand until cool enough to touch, about 15 minutes.
Meanwhile, in a large bowl, mix celery, mayonnaise, vinegar, onion, and pickle relish. Peel warm potatoes, cut into 1-inch chunks, and add to dressing; mix gently. Let stand at room temperature or chill until cool.
Add eggs, parsley, and chopped dill; mix gently and season to taste with salt and pepper. Cover and chill until cold, at least 2 hours. Garnish with dill sprigs.