Place potatoes in a saucepan, and cover with water; bring to a boil. Reduce heat; simmer 10 minutes or until tender. Drain and rinse with cold water. Cool. Cut potatoes into 1/4-inch-thick slices. Place potatoes in a large bowl; add shallots, 2 tablespoons vinegar, and parsley. Toss gently.
Combine 1/4 cup vinegar, oil, salt, and pepper in a small bowl; stir well with a whisk. Pour dressing over potato mixture; toss gently to coat. Add radishes; toss gently. Sprinkle with chives.