Cook the fettuccine according to package instructions. Reserve 1 cup of pasta water before draining.
In a large skillet, heat the olive oil over medium-high heat. Season the chicken breasts with salt and pepper, then add them to the skillet. Cook until browned and cooked through, about 5 minutes per side. Remove the chicken from the skillet and set aside.
In the same skillet, add the mushrooms and garlic. Cook until the mushrooms are browned and softened, about 5 minutes. Remove half of the mushrooms from the skillet and set aside for garnish.
Add the chicken broth to the skillet and bring to a boil. Reduce heat to low and simmer for 5 minutes. Stir in the heavy cream and Parmesan cheese. Cook for an additional 2 minutes, until the sauce has thickened.
Return the cooked chicken to the skillet and add the cooked fettuccine. Toss to coat the pasta in the sauce and heat through. If the sauce is too thick, add some of the reserved pasta water to thin it out.
Serve the creamy chicken and mushroom fettuccine garnished with the reserved mushrooms and chopped parsley.
