Heat the olive oil in a large skillet or saucepan over medium heat.
Add the finely chopped garlic and sauté until fragrant, about 1 minute; do not let it brown.
Add the canned peeled tomatoes with their puree, breaking up the tomatoes with a spoon, then stir in the halved Kalamata olives, chopped anchovy fillets, drained capers, dried oregano, and crushed red pepper.
Bring the sauce to a gentle simmer, then reduce heat to medium-low and cook until slightly thickened, about 8 minutes; season with salt and freshly ground black pepper to taste.
Meanwhile, bring a large pot of salted water to a boil and cook the spaghetti until al dente according to package directions.
Drain the pasta, reserving a splash of cooking water if desired, and return the spaghetti to the pot.
Add the puttanesca sauce and chopped parsley to the pasta and toss over low heat until the sauce coats the spaghetti, about 2–3 minutes; add a little reserved pasta water if needed to loosen the sauce.
Serve immediately with grated Parmesan cheese on the side.
