Bring the chicken stock to a boil in a large pot over high heat. Reduce the heat to medium, stir in the orzo and simmer, stirring occasionally, until the orzo is tender, about 15 minutes.
Whisk the eggs in a bowl until frothy. Whisk in the lemon juice in a steady stream. Slowly whisk 1 cup of the hot stock into the egg mixture until slightly thickened. Whisk the warmed egg mixture back into the pot. Reduce the heat to medium low and simmer gently, stirring, until thickened, about 5 minutes. Season with salt and pepper.
Transfer the soup to a terrine or divide it among soup bowls, garnish with chopped parsley and olive oil, and serve hot.