Soak the beans. Pick through the beans for any rocks. Place in a large container with 2 tablespoon salt and 10 cups water. Cover and let soak at least 8 hours, or overnight.
Cook the beans. Drain and rinse, then transfer to a 5- to 6-quart pot. Cover with 2 quarts fresh water. Add the bay leaves and half the onion. Bring to a boil over high heat, then reduce to medium. Cook about 1 hour, stirring occasionally and mashing some beans against the side of the pot to thicken.
Brown the ham. With 15 minutes left on the beans, heat 1 tablespoon oil in a large skillet over medium-high. Add the ham and cook 7–9 minutes until starting to brown. Transfer with a slotted spoon to the bean pot.
Brown the sausage. In the same skillet over medium-high, heat the remaining oil. Cook the andouille 7–9 minutes until starting to brown. Transfer to the bean pot.
Build the trinity. In the same skillet over medium, combine the celery, bell pepper, and remaining onion with 1/2 teaspoon salt. Cook 4–5 minutes until softening. Add the garlic and thyme; cook 1–2 minutes more.
Deglaze and combine. Pour the broth into the skillet to deglaze, then add everything to the bean pot. Bring to a simmer and cook about 30 minutes, until the liquid is the texture of gravy. Discard the bay leaves.
Serve. Spoon over white rice and top with parsley.