Adjust oven rack to middle position and preheat to 450°F.
Cut each avocado in half and remove the pit. Using a spoon, scrape out the center of each halved avocado so that it is large enough to accommodate an egg. Squeeze lime juice over the avocados and then place on baking sheet. Break an egg into the center of each avocado.
Bake in the oven until whites are set and yolk is runny, about 10 to 12 minutes. Remove from oven and garnish with salt, pepper, and parsley.