Heat a Dutch oven over medium-high heat. Add 2 teaspoons oil, sunflower seeds, and 1/4 teaspoon salt; sauté 2 minutes or until lightly browned. Remove from pan; set aside.
Heat pan over medium heat; add remaining 2 teaspoons oil and butter. Add leek; cook 4 minutes or until tender, stirring frequently. Add 2 1/2 cups water and next 3 ingredients (through rice); bring to a boil. Cover, reduce heat, and simmer 35 minutes. Stir in currants and bulgur; cover and simmer 10 minutes or until grains are tender. Remove from heat; stir in remaining 1/4 teaspoon salt, sunflower seeds, parsley, and pepper. Serve immediately.