To a large saucepan, combine the Parmesan rinds, carrots, celery, garlic, onions, oregano, thyme, bay leaf and salt. Add 8 cups water and bring to a simmer over medium heat. Simmer the broth for 1 hour 15 minutes, adjusting the heat to maintain a simmer and stirring occasionally to prevent sticking. Strain into a clean saucepan and keep warm over low heat.
Meanwhile, bring a large pot of well-salted water to a boil. Add the tortellini and cook for 2 minutes less than the package instructions, about 4 minutes. Drain well, then add the tortellini to the broth to finish cooking. Stir in the parsley and serve.