Bring a large pot of well-salted water to a boil. Add the pasta and cook until al dente. Drain, reserving 1 cup of the pasta water, and return the cooked pasta to the pan.
Meanwhile, prepare the walnut pesto by adding the basil, arugula, walnuts, olive oil, lemon juice, garlic, and sea salt to a food processor and processing for 1 to 2 minutes or until just shy of smooth, stopping to scrape down sides as needed.
Spoon the pesto over the pasta, and toss to coat, adding in pasta water by the tablespoon only if absolutely needed to thin the sauce. NOTE: If you add pasta water, you'll likely need to add more salt as well.
Taste and season with additional sea salt, if desired.
Divide the pasta between bowls.
Garnish with parmesan, if desired.