1. Rinse asparagus; snap off and discard tough stem ends. In a 12- by 17-inch baking pan, mix asparagus with 2 tablespoons olive oil. Sprinkle generously with salt and pepper. Transfer half the mixture to another 12- by 17-inch baking pan. Bake in a 450° regular or convection oven until asparagus is tender when pierced, 10 to 12 minutes.
2. Meanwhile, in a bowl, mix lemon juice and remaining 2 tablespoons olive oil. Add eggs, olives, capers, and parsley; mix gently to avoid mashing yolks. Season to taste with salt and pepper.
3. Mound arugula on a serving platter. Arrange asparagus on arugula and spoon egg salad over asparagus.