Bring a large pot of lightly-salted water to a boil; add the pasta and cook until al dente, about 5 minutes; drain. Melt 1 tablespoon butter in a skillet. Cook the hot dog pieces in the butter until browned. Add the hot dogs to the drained noodles with 2 tablespoons butter, parsley, and Parmesan cheese; stir until butter is melted and the hot dogs are coated.