Bring a large pan of water to the boil and cook the pasta for 8 minutes until starting to soften. Add the new potatoes and cook for 5 minutes more, then add the green beans and cook for 3-4 minutes until tender.
Drain the pasta, potatoes and beans and return to the pan. Add the pesto and Parmesan and toss. Scatter with basil leaves and serve immediately with extra Parmesan.