In a large stockpot, combine the onions, carrots, and celery. Toss in the parsley, thyme, and bay leaf and add the chicken parts or bones. Add enough cold water to cover the chicken by 3 inches. Bring to a boil over medium heat, lower the heat immediately so the stock simmers very gently. Skim the surface of any foam, scum, or fat, with a ladle, as often as necessary. Simmer, uncovered for 4 hours.

Strain into a large bowl or storage containers. Cool the liquid to room temperature before refrigerating. (The fastest way to cool containers of stock is in an ice water bath in the sink.) Refrigerate for 2 or 3 days or freeze for up to 3 months. Remove any solidified fat that rises to the surface before using.