Preheat the oven to 350 degrees. Grease a baking sheet. Bring a large saucepan of salted water to a boil over high heat. Cook the potatoes in the boiling water for 15 minutes, or until slightly tender on the outside but still firm in the center. Drain and cool slightly. In a small saucepan over moderately low heat, melt the butter.
Cut potatoes in half crosswise. Using a small sharp knife, make 6 or 7 parallel cuts 1/8 inch apart. Leaving 1/4 inch uncut at the bottom of each half. Place potatoes, cut side down, on the baking sheet, brush lightly with some melted butter, and bake for 10 minutes.
Remove the potatoes from the oven and gently insert a knife into the cuts to separate them and allow the butter to drip inside. Brush the potatoes with more butter and bake for 25 minutes more, or until the potatoes are cooked through and crisp, and the slices have fanned out. Garnish Hasselback potatoes with sprigs of flat leaf parsley, if desired and serve immediately. 4 servings. Preparation time: 20 minutes. Cooking time 35 minutes.