Cook the pasta according to the package directions. Drain, return the pasta 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.
Meanwhile, shell the peas. When the potatoes have 2 minutes left to cook, add the peas to the pot. Finish cooking, drain, and transfer to pasta pot.
Stir in the scallions, ¼ cup Parmesan, remaining 2 tablespoons oil, ¾ teaspoon salt, and ½ teaspoon pepper.
Divide among bowls and top with the remaining ¼ cup Parmesan.