Rinse the beans well. Place in a large pot and cover with at least 2 inches of cold water. Soak overnight. Rinse well.
Heat the oil in a large, deep pot over medium. When hot, add the onion, bell pepper, and celery and cook until starting to soften, 5-6 minutes. Add the garlic, Cajun seasoning,thyme, cayenne, and 1 teaspoon salt, and cook, stirring, for 1 minute, until fragrant.
Add the beans, broth, bay leaf, and parsley bundle. Stir well.
Bring to a simmer, and then reduce heat to medium-low, cover, and simmer for 2 1/2 hours.
Add the sausage and return to a simmer for 30 minutes, uncovered, stirring occasionally, until the beans are tender.
In a medium pot, combine the rice with 2 cups water and 1 teaspoon salt. Bring to a simmer, reduce the heat to low, cover, and simmer for 20 minutes, until the grains are tender.
Remove the parsley and bay leaf from the beans before serving with rice.