Season the chicken with salt, pepper, and Creole seasoning, and brown quickly in large heavy pot.
Brown the sausage, pour off fat, and reserve meats.
In same heavy pot, heat the oil and cook the flour in the oil over medium to high heat (depending on your roux-making skill), stirring constantly, until the roux reaches a dark reddish-brown color, almost the color of coffee, or milk chocolate for a Cajun-style roux.
If you want to save time, or prefer a more New Orleans-style roux, cook it to a medium, peanut-butter color over lower heat if you're nervous about burning it.
Add the vegetables and stir quickly.
This cooks the vegetables and also stops the roux from cooking further.
Continue to cook, stirring constantly, for about 4 minutes.
Add the stock, seasonings, chicken and sausage.
Bring to a boil, then simmer for about 1 hour, skimming fat off as needed.
Add the chopped onion tops and parsley, and heat for 5 minutes.