Heat the oil in a small heavy saucepan and add the necks, feet, and gizzards. Sprinkle with salt and pepper. Cook, stirring frequently, until parts are nicely browned all over. Drain.
Return the squab pieces to the saucepan and add the celery, onion, carrots, thyme, parsley, and bay leaf. Continue cooking, stirring occasionally, about 5 minutes.