Serve this packet hot off the grill, but be cautious when opening it.
Preheat a grill to medium (if you are using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 5 to 6 seconds). Cut a piece of heavy-duty foil and a piece of parchment to 12 by 18 inches. Place foil on a work surface; lay parchment on top. Toss mushrooms, garlic, parsley, thyme, and oil in a medium bowl; season with salt and pepper. Mound mushroom mixture in center of parchment, allowing a 3-inch border all around.
Hold the long sides of parchment and foil; bring together, and fold twice, creasing to seal firmly. Fold short ends twice, creasing to seal firmly.
Grill packet until you hear mushrooms and liquid sizzling, 10 to 15 minutes. Flip packet, and grill until mushrooms are soft, about 15 minutes more. (Press lightly to test tenderness, or unfold center of packet to check, but be careful of steam escaping.) Remove packet, and let stand 5 minutes before serving.