1. In a medium saucepan, bring the turkey neck, giblets, 4 cups water, 2 cups chicken stock, the celery, onion and garlic to a boil. Reduce the heat and simmer, partially covered, for 2 hours. Strain the broth. Measure out 1/2 cup. Set aside the remainder.
2. Pour the roasted-turkey pan juices into a 4-cup glass measure. Let sit until the fat rises to the top. Spoon off and discard all but 1/4 cup fat. Add all but the reserved 1/2 cup giblet broth to the pan juices. Pour in enough extra chicken stock or water to make 4 cups liquid.
3. Pour the liquid into the turkey pan. Bring to a boil over medium-high heat, scraping up any browned bits.
4. In a jar with a tight-fitting lid, shake the flour with the remaining 1/2 cup water and 1/2 cup giblet broth until smooth. Gradually whisk the mixture into the pan liquid. Boil, then simmer for 5 minutes or until thickened. Stir in the parsley, salt and pepper.