Bring a large pot of salted water to a rolling boil and cook the pasta according to package instructions until al dente. Reserve 1/4 cup of pasta water before draining.
Season the chicken pieces with salt and pepper. In a large skillet, heat 2 tablespoons of olive oil over medium-high heat. Add the chicken and cook for 6-8 minutes until golden brown and cooked through. Remove chicken from the pan and set aside.
Reduce heat to medium and add the remaining tablespoon of olive oil. Stir in the minced garlic, chopped anchovies, and red chili flakes. Sauté for 2 minutes, using a wooden spoon to break up the anchovies until they dissolve.
Add the crushed tomatoes, olives, capers, and dried oregano to the skillet. Simmer the sauce for 12 minutes, stirring occasionally, until slightly thickened.
Return the cooked chicken to the skillet and stir to coat in the sauce. Simmer for an additional 2 minutes to ensure the chicken is heated through.
Add the drained pasta to the skillet. Toss thoroughly to combine, adding a splash of the reserved pasta water if necessary to reach desired consistency. Garnish with fresh parsley and serve immediately.