Place the potatoes in a large saucepan and add enough cold water to cover. Bring to a boil.
Add 2 teaspoons salt. Reduce heat and simmer until tender, 15 to 18 minutes. Drain.
Meanwhile, heat 1 tablespoon of the oil in a large skillet over medium-high heat.
Add the sausages to the skillet and cook, turning occasionally, until browned and cooked through, 10 to 12 minutes.
In a large bowl, whisk together the cornichon brine, mustard, remaining 2 tablespoons of oil, and ¼ teaspoon each salt and pepper.
Add the potatoes, cornichons, onion, and parsley to the bowl and mix, mashing gently. Cut the sausages into large pieces and serve with the potatoes.