Place potatoes, 1 tablespoon salt, 1 tables vinegar, and 3 cups water in medium pot. Bring to boil over high heat. Cook potatoes until tender. Drain, reserving 1/2 cup of cooking liquid. Immediately toss potato pieces with 1/2 tablespoon vinegar and set aside.
Combine reserved cooking liquid with remaining vinegar, mustard, relish, sugar, and minced shallot in a large bowl. Add 5 to 6 pieces of cooked potato. Mash until smooth. Whisking constantly, add Olive Oil, capers, celery, sliced shallot, parsley, and potatoes. Season to taste with salt and pepper.