In a large saucepan, bring broth and water to a boil. Add the wild rice, salt and thyme. Reduce heat; cover and simmer for 30 minutes.
Meanwhile, in another saucepan, saute carrots, celery and onion in oil until almost tender. Add the mushrooms; saute 5 minutes longer. Add vegetables and long grain rice to wild rice. Cover and cook for 30-35 minutes or until rice is tender. Stir in parsley.