Preheat the oven to 375°F. Line a baking sheet with parchment paper.
Using a spiralizer or a grater, shred the zucchinis into thin strands. Place the shredded zucchini in a clean kitchen towel and squeeze out excess moisture.
In a large bowl, combine the shredded zucchini, cheddar cheese, garlic powder, onion powder, salt, and pepper. Mix until well combined.
Divide the zucchini mixture into four equal portions on the prepared baking sheet. Shape each portion into a nest, creating a small well in the center for the egg.
Drizzle the nests with olive oil. Bake in the preheated oven for 15 minutes.
Remove the baking sheet from the oven and carefully crack an egg into the center of each nest. Return to the oven and bake for an additional 10-12 minutes, or until the eggs are set to your liking.
Remove from the oven and let cool for a few minutes. Garnish with chopped parsley before serving.