Heat oil in a large stockpot or Dutch oven over medium-high heat. Chop neck, heart, and gizzard into 2-inch pieces; add to pan. Cook 5 minutes or until browned, stirring occasionally. Add onion, celery, and carrot; cook 4 minutes or until tender, stirring frequently. Stir in the water and broth; bring to a boil. Add thyme, peppercorns, parsley, and bay leaf. Reduce heat, and simmer 2 hours. Strain the mixture through a sieve over a large bowl; discard solids. Cool to room temperature. Cover and chill overnight. Skim the solidified fat from surface, and discard fat.
Note: Refrigerate leftover stock in an airtight container for 2-3 days, or freeze for up to 3 months.