Cook pasta according to package instructions.
Meanwhile, heat 2 tbsp. oil in a nonstick frying pan over medium-high heat. Cook squash until golden brown and tender, 16 to 17 minutes total. Remove from heat.
Make pesto: Whirl pistachios, garlic, lemon zest, and 1 cup each cheese and parsley in a food processor until finely chopped. With motor running, drizzle in remaining 1/3 cup oil; blend until creamy, 15 to 20 seconds. Season with salt and pepper.
Drain pasta, reserving 1/2 cup pasta water. Transfer pasta to pan of squash. Add pesto and lemon juice and toss gently, adding a little reserved pasta water if needed. Serve topped with more parsley and cheese.
Make ahead: Pesto, up to 2 days, chilled.