Make the giblet stock: Combine the giblets, carrots, celery, onion, parsley, and 8 cups water in a medium saucepan. Bring to a boil, reduce heat, and simmer until the liquid is reduced by half and the neck meat is very tender, 1 to 1½ hours.
Remove the giblets. Pull the meat off the neck and finely chop the remaining giblets; reserve. Strain the stock into a large measuring cup or bowl; you should have about 3 cups (add water if necessary).
Place the reserved roasting pan with the vegetables across 2 burners and cook over medium-high heat, scraping up the bits stuck to the pan, until the vegetables darken, 8 to 10 minutes. Add the vermouth and cook until syrupy, 2 to 3 minutes. Add the flour and cook, stirring, for 2 minutes. Whisk in the stock and cook until thickened, 4 to 6 minutes.