Preheat oven to 425°. Lightly coat a 7-by 11-in. pan with extra-virgin olive oil. Add mushrooms, brush all over with extra-virgin olive oil, sprinkle with salt and pepper, and bake hollow side up until browned and tender, about 10 minutes.
Pour off any accumulated juices from mushrooms, then sprinkle with Parmesan cheese.
Crack 1 egg into each mushroom and bake until eggs have set and yolks are still runny, 10 to 15 minutes.
Lightly brush olive oil over mushrooms and eggs. Season with salt and pepper to taste, along with lemon zest and parsley.