Peel or scrub the potatoes and cut them into 1-inch chunks, or halve or quarter baby potatoes so that the pieces are about 1-inch. Place them in a saucepan with salted water to cover by 2 inches. Bring the water to a boil over high heat, reduce the heat to about medium, and simmer the potatoes gently until they are tender and can be pierced with a sharp knife, about 15 minutes. Drain and let cool slightly.
Meanwhile in a large bowl, combine the olive oil, white wine (if using), Dijon mustard, onions, and salt and pepper.
Add the warm potatoes and gently toss until the potatoes are well coated with the dressing. Let cool slightly, then gently mix in the scallions and parsley. Serve at room temperature.