In a 3- to 4-quart saucepan combine potatoes, 3 Tbsp. cider vinegar, and 1 Tbsp. kosher salt. Add water (about 4 cups) to cover potatoes. Bring to boiling over medium heat. Reduce heat and simmer, uncovered, for 15 to 20 minutes, just until potatoes are tender when gently pierced with a fork. Drain well. Let stand for 2 to 3 minutes before adding dressing.
For Grainy Mustard Dressing, in a small bowl whisk together the mustard, 3 Tbsp. vinegar, sugar, and 1/2 tsp. kosher salt. In a steady stream, whisk in peanut oil until well blended.
Transfer potatoes to a mixing bowl. Add Grainy Mustard Dressing all at once. Mix gently to avoid breaking potatoes. Gently mix in sausage. Season with a generous pinch of kosher salt and a few grinds of black pepper. Add parsley and green onions. Serve warm or at room temperature.