Bring 2/3 cup vinegar to a boil over high heat in small saucepan and continue boiling until reduced by half. Remove saucepan from heat and let cool 5 minutes.
Combine mayonnaise, cooled vinegar, remaining 1 teaspoon vinegar and tarragon in medium bowl. Season, if desired, with salt and pepper. Cover and refrigerate at least 30 minutes.
Cover potatoes with water in 4-quart saucepot; bring to a boil over medium-high heat. Reduce heat and simmer 15 minutes or until potatoes are tender, but still firm. Drain and cool slightly. Cut each potato lengthwise into 8 slices.
Brush potatoes with oil and season, if desired, with salt and pepper. Grill, turning once, 4 minutes or until golden and cooked through. Arrange potatoes on serving platter, then sprinkle with parsley. Serve with dip.
Mayonnaise mixture is also delicious with hot cooked chicken fingers or fish.