Bring a large pot of salted water to a boil. Add the pasta and cook until al dente, 7 to 9 minutes or according to package instructions. Meanwhile, cook the mushrooms.
Heat 1 tablespoon of the butter and the oil in a large skillet over medium heat until shimmering. Add the mushrooms, season with salt and pepper, and sauté until browned and tender, about 5 minutes. Add the garlic, red pepper flakes if using, and remaining 2 tablespoons butter and sauté for 1 minute more.
When the pasta is ready, reserve 3/4 cup of the cooking water, then drain the pasta. Add the pasta, reserved cooking water, and cheese to the skillet. Toss over medium heat until the cheese is melted and the sauce thickens and coats the pasta, about 2 minutes. Add the parsley and toss to combine. Serve in shallow bowls with more cheese for serving.
Storage: Leftovers can be refrigerated in an airtight container for up to 4 days.