In a large pot over medium heat, heat oil. Add onion and cook, stirring, until soft, 5 minutes. Add garlic and cook until fragrant, 1 minute more. Stir in tomato paste and red pepper flakes. Add tomatoes and broth and bring to a simmer.
Add penne and cook until al dente, 8 to 10 minutes.
Add chicken, mozzarella, Parmesan, and parsley and season generously with salt and pepper. Let cheese melt, then ladle into bowls.