Preheat the oven to 425 degrees F. Pierce the potatoes in a few places with a fork and microwave until just tender, 6 to 8 minutes. Set aside to cool.
Heat 2 tablespoons olive oil in a large ovenproof nonstick skillet over medium-high heat. Add the mushrooms, season with salt and pepper and cook, stirring, until browned, 5 minutes. Add the garlic and cook until just softened, about 1 minute. Whisk 3 eggs, 1 tablespoon olive oil, the flour, milk, 3/4 teaspoon salt and a few grinds of pepper in a medium bowl. Pour over the mushrooms and sprinkle with the parsley. Transfer to the oven and bake until puffed and browned around the edges, about 20 minutes.
Meanwhile, combine the remaining 1 tablespoon olive oil and the chorizo in another large nonstick skillet; cook over medium heat, stirring occasionally, until beginning to crisp, about 5 minutes. Chop the potatoes into bite-size pieces and add to the skillet; season with salt and pepper. Cook until the potatoes are slightly crisp, 8 to 10 minutes.
Make 4 openings in the potato mixture and crack an egg into each spot; season with salt and pepper. Cook until the whites are set but the yolks are still soft, 3 to 5 minutes. Cut the Dutch baby into wedges. Serve with the potatoes, chorizo and eggs; sprinkle with parsley.