Make the Roux: In a large pot or Dutch oven, whisk together flour and oil over medium heat. Stir constantly until the mixture turns a deep brown, about 15–20 minutes.
Cook the Vegetables: Add onion, bell pepper, celery, and garlic. Cook until softened, about 5 minutes.
Add Meats: Stir in sausage and cook for 5 minutes. Add chicken and cook until lightly browned.
Simmer the Gumbo: Add diced tomatoes, chicken broth, bay leaves, thyme, paprika, cayenne (if using), salt, and pepper. Bring to a boil, then reduce heat and simmer, covered, for 45 minutes, stirring occasionally.
Add Shrimp: Stir in shrimp and simmer for 5–7 minutes, until cooked through.
Finish: Remove bay leaves and stir in chopped parsley.
Serve: Ladle gumbo over cooked white rice and serve hot.
