Cook the spaghetti in salted water to al dente according to directions on the package. Make sure to reserve about 1/2 cup of the pasta water before you drain it.
While pasta is cooking, mix the minced garlic and 1 Tablespoon water together in a small bowl. Heat 2 Tablespoons olive oil, garlic water mixture, 1 teaspoon red pepper flakes and 4 teaspoons anchovy paste over medium heat in a large skillet. Stir frequently and cook until garlic is fragrant but not browned, about 2-3 minutes. Stir in 28 ounce can diced tomatoes and simmer until slightly thickened, about 8-10 minutes.
Stir 3 Tablespoons capers, 1/2 cup pitted Kalamata olives, and 1/4 cup minced fresh parsley leaves into the sauce.
Drain the spaghetti and add it to the thickened sauce. Add the drained spaghetti to the sauce, and continue cooking and stirring for about 2-3 minutes. If the sauce becomes too thick during this time, add some of the pasta water a little bit at a time until it reaches a consistency that you like. Serve with extra minced parsley if desired.