Combine oil and butter in a large Dutch oven; cook over low heat, stirring until butter melts. Add flour, stirring until smooth. Cook over low heat 1 minute, stirring constantly.
Add celery and onion to Dutch oven; cook 10 minutes or until vegetables are tender. Stir in oysters and liquid; cook an additional 10 minutes. Gradually add water; cook over medium heat 30 minutes. Stir in milk, salt, pepper, and parsley. Serve hot.