Cook the pasta according to the package directions. Drain the pasta, return it to the pot, and toss with 4 tablespoons of the oil. While the pasta cooks, place the potatoes in a large pot, cover with cold water, and bring to a simmer. Add 1 tablespoon salt and cook until tender, about 12 minutes. Shell the peas. When the potatoes have 2 minutes left to cook, add the peas to the pot. Drain the vegetables and transfer to the pot with the pasta. Stir in the scallions, 1/4 cup Parmesan, the remaining oil, 3/4 teaspoon salt, and 1/2 teaspoon pepper. Divide among individual bowls and top with the remaining Parmesan.