1. Place chicken hearts and gizzards in a pot, add water, onion, salt and bay leaf and bring to a boil. Cover and turn heat to low and cook about 1 hour or until tender. Cool sufficiently to handle giblets. Remove giblets. Trim membranes from gizzards, and chop gizzards and hearts fine.
2. Skim off fat from broth. Measure 2 cups broth and heat to boiling. Stir in rice, giblets and celery, and bring back to a boil.
3. Cover, turn heat to low and cook 20 minutes or until rice is tender and liquid is absorbed. Stir in cheese and parsley. Heat very slowly, for a minute or two, just until cheese melts.