Set a large pot over medium heat and add the vegetable oil. When the oil is hot, add the sausage slices and cook, stirring occasionally, for 3-4 minutes, or until lightly browned. Transfer to a plate and set aside.
In the pot used for the sausage, add the onion, celery, and bell pepper. Cook for 3 minutes, until softened. Stir in the garlic and cook for an additional 30 seconds.
Add the Creole seasoning, salt, black pepper, and smoked paprika. Pour in the broth and add the potatoes. Bring to a boil, then reduce the heat and let simmer for 15 minutes, or until the potatoes are fork-tender.
Reduce the heat to low. Return the sausage to the pot. Stir in the heavy cream and cheddar cheese. Simmer for 5 minutes, until the cheese is melted.
Ladle into bowls, sprinkle with parsley, and serve hot.