Preheat the oven to 375°F (190°C).
In a skillet, heat the olive oil over medium heat. Add the minced garlic and chopped mushrooms, cooking until the mushrooms are soft and any moisture has evaporated, about 5-7 minutes.
In a mixing bowl, combine the cooked mushrooms and garlic with cream cheese, Parmesan cheese, breadcrumbs, dried thyme, salt, and pepper. Mix until well combined.
Carefully slice a pocket into each chicken breast, making sure not to cut all the way through.
Stuff each chicken breast with the mushroom mixture, using toothpicks to secure the opening if necessary.
Season the outside of the chicken breasts with salt and pepper.
Place the stuffed chicken breasts in a baking dish and bake for 25-30 minutes, or until the chicken is cooked through and reaches an internal temperature of 165°F (74°C).
Remove from the oven and let rest for 5 minutes before serving. Garnish with fresh parsley.