Boil a large pot of salted water. Cook fettuccine until just al dente. Reserve 1/2 cup pasta water, then drain.
Heat olive oil and 1 tablespoon butter in a large skillet over medium-high heat.
Sauté mushrooms in an even layer until browned on one side, about 3 minutes. Stir and cook 2 to 3 minutes more.
Lower heat to medium. Add remaining butter and garlic; cook until fragrant, about 30 seconds.
Pour in heavy cream and simmer gently, stirring, until slightly thickened, 2 to 3 minutes.
Add drained pasta to the pan. Sprinkle in Parmesan and toss, loosening with splashes of pasta water until silky.
Season with salt, pepper, and nutmeg. Toss until the sauce clings and looks glossy, 1 to 2 minutes.
Remove from heat. Fold in parsley and serve hot with extra Parmesan.