Melt butter and oil in a large skillet over medium-high heat. Add mushrooms, salt, and pepper, and cook until golden brown. Just before they're done add the garlic and thyme. Cook until fragrant, about 1 minute.
Add the wine and stir to loosen the brown bits from the bottom of the pan. Cook until almost entirely evaporated. Add the chicken broth and cream. Lower the heat to medium and simmer until reduced by half, about 5 minutes.
Stir in parsley and more salt and pepper to taste.