Place a jelly-roll pan in oven. Preheat oven to 400°.
Combine potatoes and 1 tablespoon oil in a medium bowl; toss well. Arrange potatoes in a single layer on preheated pan. Sprinkle potatoes with 1/2 teaspoon salt and 1/4 teaspoon pepper. Bake at 400° for 20 minutes; toss potatoes. Bake an additional 25 minutes or until tender, stirring every 10 minutes.
Combine remaining 3 tablespoons oil, remaining 1/4 teaspoon salt, 1/8 teaspoon pepper, arugula leaves, Parmesan cheese, pistachios, 2 teaspoons water, lemon juice, and garlic in a food processor; process until smooth. Combine arugula mixture and hot potatoes in a medium bowl; toss well.