In a saucepan, boil water, salt it, then cook the pasta according to the package instructions. When the pasta is cooked, reserve ½ cup of the pasta water and drain the pasta.
Meanwhile, in a skillet, heat oil and add the chicken. Season with salt and pepper and cook the chicken for 4-5 minutes. Remove the fully cooked chicken onto a plate and set aside.
Add heavy cream and parmesan, and whisk or stir until you get a smooth sauce. Add up to half of the pasta water if needed to loosen the sauce and whisk again.
Add the pasta and return the chicken to the skillet, and gently mix everything together until well combined. Garnish with extra parmesan cheese and fresh chopped parsley.