In a large saucepan, add the potatoes and enough water to cover by 1 inch. Add a pinch of salt and bring to a boil over high heat. Reduce the heat to medium and simmer until softened, about 20 minutes. Drain, transfer to a bowl and cover to keep warm.
Meanwhile, prick the sausages all over with a sharp paring knife. Place in a large skillet with enough water to reach a quarter of the way up the sausages. Cook over medium-high heat, turning often, until the water has evaporated, 15 to 20 minutes. Continue cooking, turning with tongs, until the sausages are browned all over, 5 to 7 minutes.
In a small bowl, whisk together the vinegar, mustard, onion, 1/2 teaspoon salt and the pepper. Slowly pour in the olive oil, whisking until the sauce is thick and emulsified. Pour over the potatoes and stir gently to coat.
Spoon the potato salad onto a platter and sprinkle the parsley on top. Serve the sausages on the potato salad.