Begin by bringing a large pot of salted water to a boil. Add the pasta and cook according to package instructions until al dente. Reserve about half a cup of pasta water before draining.
In a large skillet over medium heat, melt the butter and olive oil. Add the minced garlic and sauté until fragrant, about one minute. Stir in the gochujang until well combined.
Pour in the heavy cream and mix thoroughly with the gochujang sauce until smooth. Gradually add in the reserved pasta water as needed to reach desired sauce consistency.
Add the shredded cheese into the sauce, stirring continuously until it melts completely. If desired, sprinkle in the black pepper.
Carefully add the drained pasta into the skillet with the sauce, tossing until each piece is thoroughly coated.
Plate the pasta and sprinkle with chopped parsley before serving hot.