Place turkey carcass in a stockpot, cover with water, and bring to a boil. Reduce heat to low, cover, and simmer for 1 hour. When cool enough to handle, strain stock into a large measuring cup, and discard bones. If using oysters, strain oyster liquor into stock. If necessary, add water to measure at least 8 cups liquid. Set aside.
In a large heavy stockpot, heat oil over medium-high heat. Add flour, and stir constantly until roux begins to brown. Reduce heat to medium, and cook, stirring constantly, until roux becomes the color of peanut butter.
Add onion, green onion, and celery; simmer over low heat until translucent. Add garlic, and cook 1 minute more. Add 8 cups stock (or more if you prefer a thinner gumbo).
Add bay leaves, thyme, salt, Creole seasoning, pepper, and andouille sausage; cover and simmer for 30 minutes. Add turkey meat and oysters, if using, and cook until oysters curl, 1 to 2 minutes. Remove bay leaves, and adjust seasonings, if necessary. Add parsley, and serve in bowls over rice. Garnish with parsley and thyme, if desired.