Place the potatoes in a large saucepan and add enough cold water to cover by 1-inch. Season the water generously with salt. Bring to a boil over medium-high heat. Once at a boil, reduce the heat to a simmer and cook until potatoes are tender, about 10 minutes.
Drain the potatoes and transfer to a large bowl. Add the vinegar and, using a rubber spatula, toss gently to combine.
Toss the potatoes with the celery, onions, mayonnaise, pickled relish, whole grain mustard, chives and parsley and taste for seasoning. Cover with plastic wrap and refrigerate until chilled and to let the flavors meld, about 30 minutes.