In a small bowl soak the onions in cold water for 15 minutes. Drain.
In a medium saucepan cover the potatoes and 2 tablespoons salt with cold water by about 2 inches. Bring to a boil, lower the heat, and simmer until fork tender, about 6 to 8 minutes. Drain, transfer potatoes to a large bowl, and mix with the onions and celery. Set aside to cool.
In a small bowl, whisk together the remaining 1 teaspoon salt, mayonnaise, parsley, relish, mustard, and lemon juice.
When the potatoes have cooled, carefully fold them with the mayonnaise mixture. Season with pepper, to taste. Transfer to a serving bowl and serve.
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