Cover potatoes with water in a medium saucepan. Add 2 tablespoons salt. Bring to a boil; reduce heat, and simmer until tender, 7 to 9 minutes. Drain potatoes, reserving 1 tablespoon cooking water. Let potatoes cool slightly.
Make the vinaigrette: Whisk together vinegar, mustard, and reserved cooking water in a small bowl. Gradually add oil, whisking until emulsified. Transfer potatoes to a medium bowl, and drizzle with vinaigrette. Add onion, cornichons, capers, parsley, remaining 1 1/2 teaspoons salt, and the pepper. Using a large spatula, gently stir to combine.