In a large Dutch oven or heavy-bottomed pot over medium-high heat, cook the ground pork sausage, breaking it apart with a spoon, until browned and cooked through, 5-7 minutes. Transfer to a plate and set aside.
Add the sliced andouille sausage to the pot and cook until browned on the edges, about 3 minutes. Transfer to the plate with the pork sausage.
In the same pot, reduce heat to medium and melt the butter. Add onion, celery, and carrot. Cook until softened, 5-6 minutes.
Stir in garlic, Cajun seasoning, and cayenne; cook for one more minute.
Add potatoes and chicken broth. Bring to a boil, then reduce heat and simmer until potatoes are fork-tender, 15-20 minutes.
Return the sausage mixture to the pot and add heavy cream and shredded cheese. Let the soup simmer for an additional 5 minutes, or until everything is heated through and the cheese is melted.
Garnish with parsley. Serve warm.