Heat the olive oil in a large skillet set over medium heat. Add the garlic and cook 2-3 minutes, or until soft and fragrant. Add the mushrooms and cook 4-5 minutes, or until they begin to soften. Stir in the wine; cook until evaporated, about 2 minutes. Add a slice of mozzarella and cook until melted. Stir in the parsley and season with salt and pepper.
Butter each slice of bread. Flip 2 slices of bread over (so the buttered side is down) and top each with one slice each of mozzarella and cheddar. Top with half of the mushroom mixtures and cover with the remaining cheese. Top with the remaining bread (buttered side up).
Cook sandwiches covered over medium heat for 4 minutes on each side, or until bread is golden brown and cheese is melted.