Cook the shells in salted boiling water until al dente. Reserve about ½ cup pasta water, then drain.
Melt the butter in a large skillet over medium-high heat. Add the ground beef and onion and cook until the beef is deeply browned and the onion is tender.
Stir in the garlic, Italian seasoning, smoked paprika, garlic powder, black pepper, and salt. Cook for about 30 seconds until fragrant.
Lower the heat and add the heavy cream, milk, and cream cheese. Stir until smooth and slightly thickened.
Remove from the heat and stir in ¾ cup mozzarella, the provolone, cheddar, and ½ cup Parmesan until fully melted into a thick four-cheese sauce.
Fold in the shells until the cheese sauce fills the shells and coats the outside. Add a little reserved pasta water only if needed. Finish with the remaining mozzarella and Parmesan, cover briefly to melt, then scatter with parsley.