Bring 6 cups water, potatoes, and salt to boil in large saucepan over medium-high heat. Reduce heat to medium-low and simmer until potatoes are just tender when pierced with knife, 10 to 15 minutes.
Reserve 1/4 cup cooking water. Drain potatoes and return them to pan. Combine tapenade, lemon juice, and 2 tablespoons cooking water in bowl. Add tapenade mixture to potatoes and fold gently to incorporate. Add remaining 2 tablespoons cooking water as needed to adjust consistency. Transfer potatoes to serving bowl, sprinkle with parsley, drizzle with oil, and serve.
Notes
NOTE FROM THE TEST KITCHEN: Use small red potatoes measuring about 1 1/2 inches in diameter.