Melt the butter in a large pot or Dutch oven over medium heat. Add the flour and stir constantly with a wooden spoon or spatula to make a roux. Cook the roux, stirring constantly, until it turns a deep brown color, about 20-25 minutes.
Add the chopped onion, bell pepper, celery, and garlic to the pot and stir to combine. Cook, stirring occasionally, until the vegetables are tender, about 10-15 minutes.
Add the cayenne pepper, paprika, thyme, bay leaves, salt, and black pepper to the pot and stir to combine.
Gradually pour in the seafood or chicken stock, stirring constantly to avoid lumps. Boil the mixture, by reducing the heat to low and let it simmer for 20-30 minutes by stirring occasionally.
Add the cooked crawfish tails to the pot and stir to combine. Simmer for an additional 5-10 minutes, or until the crawfish are heated through.
Remove the bay leaves and discard them. Stir in the chopped parsley.
Serve the etouffee hot over cooked white rice.