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 anchovies, 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 10 minutes, stirring occasionally.
Stir in the Kalamata olives and capers. Simmer for an additional 5 minutes to blend the flavors.
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 to loosen the sauce as needed.
Remove from heat and stir in the chopped parsley.
Divide the pasta among serving plates and serve immediately.