Make the roux: In a large Dutch oven or heavy-bottomed pot, whisk together the flour and oil. Cook over medium heat, stirring constantly, until the mixture turns a dark chocolate brown. This can take 30-60 minutes. Be patient and do not burn it.
Add the holy trinity: Add the chopped onion, bell pepper, and celery to the roux. Cook, stirring, until softened, about 8-10 minutes. Add the minced garlic and cook for 1 minute more until fragrant.
Add broth and seasonings: Gradually whisk in the chicken broth until smooth. Add the sliced andouille sausage, chicken thighs, bay leaves, thyme, and cayenne pepper.
Simmer: Bring the gumbo to a simmer, then reduce the heat to low, cover, and cook for at least 1.5 to 2 hours, stirring occasionally. The longer it simmers, the richer the flavor.
Season and finish: Remove the bay leaves. Season the gumbo with salt and black pepper to your taste. Stir in the fresh parsley and green onions.