Heat the olive oil in a large skillet over medium heat. Season the chicken breasts with salt and pepper on both sides.
Add the chicken breasts to the skillet and cook for about 6-7 minutes on each side, until golden brown and cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, melt the butter. Add the sliced mushrooms and cook for about 5 minutes, stirring occasionally, until they are tender and have released their moisture.
Add the minced garlic to the skillet and sauté for an additional minute until fragrant.
Pour in the chicken broth and scrape any brown bits from the bottom of the skillet for added flavor. Let it simmer for 2-3 minutes.
Reduce the heat to low, slowly stir in the heavy cream, thyme, and parsley. Let the sauce simmer for another 3-5 minutes, until it thickens slightly. Season with additional salt and pepper if needed.
Return the chicken to the skillet, spoon some sauce over the chicken, and let it simmer for an additional 2 minutes to heat through.
Serve the chicken drizzled with mushroom cream sauce, garnished with fresh parsley.