Heat the oil in a Dutch oven over medium heat. Whisk in the flour and reduce heat to medium-low. Stir constantly for 35–40 minutes until the roux is a deep chocolate-brown.
While the roux cooks (or in a separate pan), brown the andouille sausage and chicken thigh pieces until golden on all sides, about 6–8 minutes. Set aside.
Once the roux is dark and ready, immediately stir in the onion, celery, and bell pepper. Cook 5 minutes, then add the garlic for 1 more minute.
Slowly whisk in the chicken broth. Add the diced tomatoes, bay leaves, thyme, smoked paprika, cayenne, and Cajun seasoning. Bring to a simmer, then return the sausage and chicken to the pot.
Simmer uncovered for 30–35 minutes, stirring occasionally. Stir in the sliced okra during the last 10 minutes.
Add the shrimp and cook 5–6 minutes, just until pink and curled. Remove from heat immediately. Taste and adjust seasoning.
Let rest 10 minutes, then serve hot over white rice, garnished with green onions and parsley.