Bring a large pot of salted water to a boil and cook the pasta according to package directions until al dente; reserve about 1 cup of the pasta cooking water, then drain the pasta.
While the pasta cooks, add the arugula, olive oil, garlic, walnuts, grated Parmesan, salt, and pepper to a blender.
Pulse the mixture until smooth, stopping to scrape down the sides as needed; if the pesto is too thick, add a few tablespoons of the reserved pasta water and blend until you reach the desired consistency.
Toss the drained pasta with the arugula-walnut pesto, adding more reserved pasta water a little at a time to loosen the sauce and help it coat the pasta.
Serve immediately and top with additional grated Parmesan if desired.