Put potatoes and eggs in a large (5- to 6-quart) saucepan of cold water. Add 2 teaspoons salt and bring to a simmer over high heat. Cook eggs 12 minutes; then remove and place in a bowl of ice water to cool. Continue cooking potatoes until you can poke them easily with a sharp knife, 2–3 minutes more. Drain potatoes in a colander and let cool, then cut into 1-inch chunks and sprinkle with half of vinegar. Peel and chop eggs.
In a small bowl, stir together eggs, remaining vinegar, oil, scallions, celery, pickles, mayonnaise, and parsley. Add pepper and remaining teaspoon salt.
Taste and adjust seasoning if needed, then toss with cooled potatoes. Refrigerate until ready to serve.