Place potatoes in a large saucepan and cover with cold water. Lightly salt and bring to a boil. Reduce heat to medium; cook 15 to 20 minutes, until fork-tender. Drain and place in a serving bowl.
Meanwhile, heat 2 tbsp of the olive oil in a large skillet over medium heat. Add leeks, shallots and garlic; cook 10 minutes, until leeks are tender. Stir occasionally.
Toss potatoes with leek mixture. Whisk together remaining 4 tbsp olive oil, the vinegar, mustard, salt and pepper. Fold into potatoes; gently stir in parsley. Serve warm or at room temperature.