Cook potatoes in large pot of boiling salted water until tender when pierced with fork, about 15 minutes. Drain. Cool potatoes until lukewarm. Cut potatoes in half. Place in large bowl. Sprinkle wine over potatoes. Heat 2 teaspoons oil in large nonstick skillet over medium heat. Add onions and saute until tender, about 5 minutes. Add vinegar, mustard and sugar to skillet and stir to blend. Pour over potatoes and toss to coat. Add radishes, green onions, parsley and remaining 1 teaspoon olive oil and toss to blend. Season salad to taste with salt and pepper. Mound salad on platter. Surround with cucumber slices and serve.