Place the pork shoulder and pork liver in a large pot and cover with water. Bring to a boil, then reduce heat and simmer for about 1 hour until the meat is tender. Reserve 1 cup of the cooking liquid before draining.
Grind the cooked pork and liver using a meat grinder or pulse in a food processor until finely chopped but not pureed.
In a skillet over medium heat, sauté the diced onion and bell pepper for 3 to 4 minutes until softened. Add the minced garlic and cook for 30 seconds.
In a large bowl, combine the ground meat mixture, sautéed vegetables, salt, black pepper, cayenne, thyme, paprika, cooked rice, green onions, and parsley. Mix well, adding a little reserved cooking liquid if needed to keep the mixture moist but not wet.
Stuff the mixture loosely into the cleaned sausage casings and tie into links.
To cook, place the boudin in a pot of gently simmering water for 10 to 15 minutes until heated through. Do not boil.
Serve warm for a classic Southern-style sausage full of rich, savory flavor.
