Cook the fettuccine or spaghetti according to package instructions in a large pot of salted boiling water until al dente. Drain and set aside, reserving 1/2 cup of pasta water.
In a large skillet, heat the olive oil over medium heat. Add the chopped onion and sauté for about 3-4 minutes, until translucent.
Add the minced garlic and sliced mushrooms to the skillet. Cook for 5-7 minutes, stirring occasionally, until the mushrooms are browned and have released their moisture.
Stir in the dried thyme, salt, and black pepper. Cook for an additional minute to combine the flavors.
Reduce the heat to low and pour in the heavy cream, stirring to combine. Allow the sauce to simmer for 2-3 minutes until slightly thickened.
Add the cooked pasta to the skillet, tossing to coat in the creamy mushroom sauce. If the sauce is too thick, add reserved pasta water a little at a time until desired consistency is reached.
Stir in the grated Parmesan cheese until melted and well combined.
Serve immediately, garnished with fresh parsley.