Bring a large pot of salted water to a boil. Add the spaghetti and cook according to package instructions until al dente. Drain, reserving 1/2 cup of pasta water.
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 and stir to combine. Simmer the sauce for 10 minutes, stirring occasionally.
Stir in the halved Kalamata olives and rinsed capers. Continue to simmer the sauce for an additional 5 minutes.
Season the sauce with salt and freshly ground black pepper to taste.
Add the drained spaghetti to the skillet with the sauce. Toss to combine, adding reserved pasta water a little at a time if the sauce is too thick.
Remove from heat and stir in the chopped fresh parsley.
Divide the pasta among serving plates and, if desired, sprinkle with grated Parmesan cheese before serving.