Soak the onions in a small bowl of cold water for 15 minutes. Drain.
Put the potatoes in a medium saucepan with cold water to cover by about 2 inches and season generously with salt. Bring to a boil, lower the heat, and simmer until fork tender, about 8 minutes. Drain, transfer the potatoes to a large bowl, and mix with the onions and celery. Set aside to cool.
Place the eggs in a medium saucepan and cover with cold water. Bring to a boil over medium-high heat. Reduce the heat to a simmer and continue to cook for 7 minutes. Drain and transfer the eggs to a bowl of ice water until cool, about 5 minutes. Peel and roughly chop the eggs.
Meanwhile, in a small bowl, whisk together the mayonnaise, parsley, relish, mustard, lemon juice, and 1 teaspoon salt. Carefully mix the mayonnaise into the potatoes with a rubber spatula. Add the eggs, taking care not to break them up too much. Season the salad with pepper to taste. Serve.
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