Start with the roux - heat oil in a large heavy-bottomed pot over medium heat. Gradually whisk in flour. Cook, stirring constantly, until dark brown (about 20-25 minutes). Don't rush this step!
Add onion, bell peppers, and celery (the "holy trinity") to the roux. Cook until vegetables soften, about 5-7 minutes. Add garlic and cook for another minute.
Stir in sliced andouille sausage and chicken. Cook until chicken begins to turn white, about 5 minutes.
Pour in chicken stock gradually, stirring constantly to prevent lumps. Add bay leaves, Cajun seasoning, and thyme.
Bring to a boil, then reduce heat and simmer for 45 minutes, stirring occasionally.
Add okra and cook for an additional 10-15 minutes until tender.
Serve hot over rice, garnished with fresh parsley.