Warm a 15-quart (14 L) heavy-bottomed pot over medium heat for 3 minutes, then add the butter. When it has melted, add the onions and cook, stirring often, until soft and translucent, about 35 minutes.
Add the tomatoes, stir, and reduce the heat to low. Cover and let the vegetables smother together for 30 minutes or until the tomatoes are completely broken down. Add the bell pepper, celery, and garlic and stir. Cover and cook, stirring occasionally, for 45 minutes more until the vegetables are very soft, with no bite remaining. Add the flour and whisk until completely incorporated.
Add the stock and bring to a boil over high heat, then reduce the heat to low and simmer, stirring occasionally, until the mixture thickens enough to coat the back of a spoon, about 45 minutes. If you have crawfish heads from a boil, you can add them while simmering down the stock for extra flavor.
Put the crawfish tails and any fat from the bag in a large bowl and season with the salt, black pepper, cayenne, and hot sauce. Add the tails to the pot and stir to combine. Cook over low heat until the crawfish is heated through, about 5 minutes. Taste and adjust the seasoning.
Serve the étouffée over rice, garnished with the parsley and green onions.