Place the potatoes and 2 heaping tablespoons of salt in a large pot of water. Bring the water to a boil and cook until the potatoes are fork-tender, 10 to 15 minutes. Drain, then chop into 1/2-inch pieces.
Meanwhile, put the eggs in a large saucepan and cover with water. Bring to a boil, then reduce to a simmer and cook until hard-boiled, about 10 minutes. Drain and run under cold water to cool. Peel the eggs and chop.
Combine the potatoes, mayonnaise, mustard, celery, parsley, chives, onion and a pinch each of salt and pepper in a large bowl. Toss to coat the potatoes with the dressing, then add the hard-boiled eggs. Gently stir the potato salad, taking care not to mash the eggs.