In a large pot of boiling, salted water, cook the pasta according to package directions until al dente; drain, reserving 1 cup pasta cooking water.
Meanwhile, in a large nonstick skillet, heat the olive oil over medium heat; add the onion and cook until softened, about 8 minutes. Add the garlic and cook for 1 minute more. Increase the heat to medium-high, crumble in the turkey, season with salt and black pepper and cook until the turkey is just no longer pink, about 5 minutes. Stir in the tomatoes and crushed red pepper and cook, stirring occasionally, until thickened, about 15 minutes. Stir in the parsley and cheese.
Toss the pasta with the sauce and 1/2 cup reserved pasta cooking water, adding more water to thin the consistency, if desired. Serve immediately with more cheese.