Place potatoes in a saucepan; cover with cold salted water. Bring to a boil, reduce heat to low, and simmer until tender, 4 to 6 minutes. Drain.
Gently toss hot potatoes with pickle brine and vinegar in a bowl. Let cool 15 minutes, stirring occasionally.
Stir together celery, parsley, mayonnaise, and pickles in a separate bowl. Add potatoes and any remaining brine mixture and gently toss to combine. Season with salt and pepper. Serve immediately, or chill up to 2 days.