Bring a large pot of salted water to a boil. Add the potatoes and cook until fork-tender, about 20 minutes; drain. Let sit until cool enough to handle, then peel and cut into 1/2-inch pieces.
Transfer the potatoes to a large bowl. Sprinkle with the vinegar and stir gently with a rubber spatula to combine. Stir in the celery, onion, relish and most of the parsley. Fold in the mayonnaise and season with salt and pepper.
Transfer the salad to a serving bowl and sprinkle with the remaining parsley.