Cook pasta in large pot of boiling salted water until just tender but still firm to bite, stirring occasionally. Drain, reserving 1/2 cup pasta cooking liquid. Return pasta to pot.
Meanwhile, bring cream to simmer in large skillet over medium-high heat. Add peas and simmer just until heated through, 1 to 2 minutes. Add 2 1/4 cups cheese and stir until melted and sauce thickens slightly, about 1 minute. Stir in mint and 1/4 cup parsley. Pour sauce over pasta and toss to coat, adding pasta cooking liquid by tablespoonfuls if dry. Season to taste with salt and pepper. Transfer to bowl. Sprinkle with remaining parsley. Serve, passing additional Parmesan alongside.