Place onion, carrot, bay leaf and chicken broth in a large pot over medium/high heat. Bring just to a boil and then reduce to a simmer. Cook 20 minutes, or until vegetables are just soft.
In a bowl whisk eggs well. Whisk in lemon juice. While continuously whisking egg and lemon juice, add in 1 ladle full of the hot soup mixture. Whisk 1 minute and add another ladle of hot liquid. Again, whisk for 1 minute.
Reduce soup heat to low and stir whisked egg, lemon juice and hot broth mixture into soup. Season with salt and pepper. Cook, just 5 minutes on low. Stir in cooked rice and parsley and serve.