Layer the onions and salt in a bowl. Pour over the vinegar. Set aside for 30 minutes to draw out the bitter juices.
Meanwhile, place the potatoes in a pot of cold salted water. Bring to a boil, and 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 the still-warm cut potatoes in a medium size bowl. Drain the onions from the vinegar, reserving a little vinegar as dressing. Stir in olive oil and flat-leaf parsley.
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