Pulse parsley with cheese, pine nuts, garlic, and 1/2 teaspoon each of salt and pepper in a food processor until finely chopped. With motor running, add 1/2 cup oil in a slow stream, blending until incorporated and almost smooth.
Cover potatoes with water in a medium pot and season well with salt, then boil until tender, 10 to 12 minutes. Drain in a colander.
Heat remaining tablespoon oil in a wide medium pot over medium-high heat until it shimmers. Sauté sugar snaps until crisp-tender, about 4 minutes. Remove from heat and add potatoes and pesto, stirring gently to coat. Transfer to a serving dish to cool quickly (so pesto doesn't separate).
Cooks' note: Pesto can be made 2 days ahead and chilled, its surface covered with plastic wrap.