In a medium bowl, whisk together eggs, cheese, parsley, pepper and 1 teaspoon salt. Cover mixture and refrigerate.
Pour water and remaining teaspoon salt into a large saucepan and bring to a boil. Stir in the rice and reduce heat to low. Cook rice until water is almost absorbed, about 10-12 minutes, stirring frequently. Remove saucepan from heat and slowly pour in egg mixture, stirring rapidly to prevent egg from scrambling. Allow rice mixture to cool in refrigerator for 1 hour.
Pour breadcrumbs into a pile on one end of a cutting board. Dampen hands and roll rice mixture into 1-inch balls. Coat each one with breadcrumbs.
In a small deep skillet, heat olive oil to 350F (make sure the oil will cover the rice balls). Fry rice balls 6 at a time, turning as needed to ensure even browning. Drain on paper towels and serve warm.