Layer onions and salt in a bowl. Pour vinegar over onions. Set aside for 30 minutes to draw out the bitter juices.
Meanwhile, place potatoes in a pot of cold salted water. Cook until fork tender, 15 to 20 minutes. Drain and cut potatoes in half. Cut each half into 1/4-inch slices to form half-moons.
Place still-warm cut potatoes in a medium-size bowl. Drain onions from vinegar, reserving a little vinegar as dressing. Stir in olive oil and flat-leaf parsley.