Cook the pasta in a large pot of salted boiling water according to package directions until al dente. Reserve ½ cup of the pasta cooking water before draining.
Meanwhile, heat the olive oil in a large skillet over medium-high heat. Add the sliced mushrooms and cook, stirring occasionally, until the mushrooms release their liquid and begin to brown, about 5-7 minutes.
Add the minced garlic, thyme, salt, and pepper to the mushrooms. Cook for 1-2 minutes until fragrant.
Pour in the white wine, if using, and simmer until mostly evaporated, scraping up any browned bits from the bottom of the pan.
Reduce the heat to medium-low and stir in the heavy cream. Simmer gently for 2-3 minutes until slightly thickened.
Add the cooked pasta to the skillet with the mushroom sauce. Toss well to coat, adding some of the reserved pasta water if needed to loosen the sauce.
Remove from heat and stir in the grated Parmesan cheese until melted. Taste and adjust seasoning as needed.
Serve the creamy mushroom pasta hot, garnished with chopped fresh parsley and extra Parmesan cheese.
