Preheat oven to 400°. Grease a baking sheet with cooking spray. Remove stems from mushrooms and finely chop. Arrange caps on prepared sheet.

In a medium skillet over medium heat, melt butter. Add chopped mushroom stems and cook, stirring frequently, until most of the moisture is evaporated, about 5 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more. Add bread crumbs and cook, stirring frequently, until slightly toasted, about 3 minutes; season with salt and pepper. Let cool slightly.

Transfer bread crumb mixture to a large bowl. Add Parmesan, cream cheese, parsley, and thyme; season with salt and pepper and stir to combine. Fill mushroom caps with filling and sprinkle with more Parmesan.

Bake stuffed mushrooms until softened and the tops are golden, about 20 minutes.

Transfer stuffed mushrooms to a platter. Top with parsley.