Place potatoes in a large pot with enough water to cover by several inches. Bring to a boil over high heat, add 1 tablespoon salt, and lower to a gentle boil. Cook until potatoes are tender when pierced with a knife, about 25 minutes. Drain into a colander. Using paper towels or gloves to protect your hands, peel potatoes, and cut into 1-inch dice while still hot. Drizzle with vinegar; set aside.
Place eggs in a small pan with enough water to cover by 1 inch, and place over medium-high heat. When water comes to a boil, turn off heat, cover, and let stand for 13 minutes. Drain, and place in a bowl with cold water to cover. When cold, peel eggs. Chop 2 into 1/4-inch dice. Slice the third egg into 1/4-inch-thick rounds, and set aside for garnish.
Combine diced eggs, mayonnaise, celery seed, mustard, 2 teaspoons salt, and black pepper in a large bowl, and whisk to combine. Add reserved potatoes to mayonnaise mixture. Add celery, red pepper, onion, cornichons, scallions, and chopped parsley. Stir to combine. Chill for 30 minutes before serving. Garnish with paprika and reserved hard-boiled egg rounds.