In a small bowl soak the onions in cold water, for 15 minutes. Drain.
In a medium saucepan cover the potatoes with cold water by about 2 inches. Stir in 2 tablespoons of the salt. Bring to a boil, lower the heat, and simmer until fork tender, about 6 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, and mustard.
When the potatoes have cooled, carefully fold them with the mayonnaise mixture. Gently mix in the eggs and season with pepper to taste. Transfer to a serving bowl and serve.
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