In a saucepan bring water to a boil. Add rice and stir. Reduce heat, cover and simmer for 20 minutes. Once cooked, spoon rice into a large bowl and allow to cool until it is cool enough to handle.
Meanwhile, beat eggs together with 1/4 teaspoon salt. Brush the bottom of a skillet with vegetable oil, and place over medium heat. Once hot, pour in 1/6 of the egg mixture and spread evenly over the bottom of the pan. Cook until the egg has just firmed, then carefully flip over and cook for a few more seconds to firm the other side. Repeat with remaining egg to create 6 thin sheets.
Stir together vinegar, sugar, and 1 teaspoon salt in a small bowl. Microwave for a few seconds until the mixture is hot; stir until the sugar has dissolved. Fold vinegar and sesame seeds into warm rice.
To assemble, place a large spoonful of rice into the center of each egg sheet. Fold into a square, and tie with a sprig of Italian parsley to secure.