Cook the pasta according to the package directions; drain and return it to the pot.
Meanwhile, heat the oil in a large skillet over medium-high heat. Add the mushrooms, onion, garlic, ½ teaspoon salt, and ¼ teaspoon pepper and cook, stirring frequently, until the onion is soft, 5 to 7 minutes.
Add the beef to the skillet and cook, breaking it up with a spoon, until browned, 3 to 5 minutes.
Add the tomato paste and cook, stirring, until slightly darkened, about 1 minute. Add the wine and cook, stirring, until nearly evaporated, about 1 minute. Add the tomatoes (and their juices) and simmer until the liquid is slightly thickened, 4 to 5 minutes.
Add the sauce to the pasta and toss to combine. Serve sprinkled with the Parmesan.