In a large saucepan, cover potatoes with water by several inches. Bring to a boil, then add 1 tablespoon salt. Reduce heat and gently boil until potatoes are tender when pierced with the tip of a sharp knife, about 25 minutes. Drain. Peel potatoes while still hot, using paper towels to protect hands; cut into 1-inch pieces. Transfer potatoes to a bowl and drizzle with vinegar; let cool.
Place eggs in a small saucepan; fill with enough cold water to cover by 1 inch. Bring to a boil; turn off heat. Cover; let stand 11 minutes. Transfer to a bowl and cover with cold water; let cool and peel. Cut 2 eggs into 1/4-inch dice. Slice remaining egg into 1/4-inch-thick rounds; reserve for garnish.
Combine diced eggs, mayonnaise, celery seeds, and dry mustard in a large bowl; season with salt and pepper, and whisk to combine. Stir in potatoes, celery, onion, cornichons, scallions, and parsley. Refrigerate at least 30 minutes or up to 1 day. Just before serving, garnish with paprika and egg rounds.
Reprinted with permission from Martha's American Food by Martha Stewart, Clarkson Potter copyright (c) 2012