Preheat oven to 400 degrees. Lightly oil a rimmed baking sheet.
In a food processor, pulse bread and garlic until fine crumbs form; set 1/2 cup aside. To food processor, add goat cheese, parsley, and red-pepper flakes. Season with salt, and pulse filling until combined.
Spoon filling into each mushroom, and roll filled side in reserved breadcrumbs. Place on prepared baking sheet; bake until mushrooms are tender and lightly browned, 15 to 20 minutes.