Bring a large pot of salted water to a boil. Add the spaghetti and cook according to package instructions until al dente. Reserve 1 cup of pasta cooking water, then drain the pasta.
While the pasta cooks, heat the olive oil in a large skillet over medium heat.
Add the minced garlic, chopped anchovy fillets, and red pepper flakes to the skillet. Cook, stirring frequently, until the garlic is fragrant and the anchovies have dissolved, about 2 minutes.
Add the crushed tomatoes to the skillet. Stir to combine and bring to a simmer. Cook for 8-10 minutes, stirring occasionally, until the sauce thickens slightly.
Stir in the Kalamata olives and capers. Simmer for an additional 2 minutes to heat through.
Season the sauce with salt and freshly ground black pepper to taste. Be cautious with salt since the anchovies, olives, and capers are salty.
Add the drained pasta to the skillet with the sauce. Toss to coat the pasta evenly, adding reserved pasta water a little at a time if needed to loosen the sauce.
Remove from heat and stir in the chopped parsley.
Divide the pasta among serving plates. Sprinkle with grated Parmesan cheese if desired and serve immediately.