Heat the olive oil in a large pot over medium heat. Add the onion, carrots, and celery with a pinch of salt. Cook, stirring occasionally, until softened, about 6 to 8 minutes. Add the garlic and cook 1 more minute.

Pour in the chicken broth and add the Parmesan rind if using. Bring to a boil, then add the pastina. Reduce the heat to a simmer and cook until the pasta is just tender, about 8 minutes, stirring occasionally.

While the soup simmers, whisk together the eggs, ½ cup Parmesan, and 1 tablespoon of the lemon juice. Turn the heat under the soup to the lowest setting. Slowly drizzle the egg mixture into the hot soup, stirring constantly with a fork, until thin strands form.

Season to taste with salt, pepper, and more lemon juice. Remove the Parmesan rind. Stir in most of the parsley, then ladle the soup into bowls and top with extra Parmesan and the remaining parsley. Serve immediately.