Bring a large pot of water to a boil and generously season with salt. Add the gnocchi and cook for 1 minute less than the package directions, or just until the gnocchi begins to rise to the top. Use a slotted spoon to remove the gnocchi to a colander.
Put the pot back on the heat and return the water to a boil. Add the green beans and peas and cook until tender, 2 to 3 minutes. Drain and let cool.
Combine the pesto and mayo in a large bowl. Stir in the Parmesan and the cooled beans and peas. Season with salt and pepper. Gently fold in the gnocchi. Garnish with more Parmesan, if desired.