In a large pot of salted water over medium-high heat, add the potatoes. Bring to a boil and cook until fork-tender, 15 to 20 minutes. Drain the potatoes in a colander. Slice the potatoes using the thick blade of a food processor and set aside. Slice the onions using the thin blade of a food processor and set aside. Finally chop the parsley using the thin blade. Place one-third of the potatoes into a large resealable food storage container. Drizzle one-third of the oil over them, then one-third of the vinegar. Top the potatoes with one-third each of the onions, salt, pepper, mustard, paprika, and parsley, in that order. Repeat the process twice. Cool to room temperature, then cover and refrigerate at least overnight.
Turn the container over for several hours so the juices permeate the potatoes. When ready to serve, bring the potato salad to room temperature and serve with a slotted spoon.