Cook the pasta according to the package directions, adding the green beans and peas during the last 3 minutes of cooking. Reserve 1 cup of the cooking water; drain the pasta and vegetables and return them to the pot.
Meanwhile, heat the oil in a large skillet over medium-high heat. Add the sausage and cook, turning occasionally, until browned, 6 to 8 minutes.
Add the sausage, pesto, Parmesan, and ½ cup of the reserved cooking water to the pasta and vegetables and toss to combine (add more cooking water as needed to loosen the sauce).