Melt 1/2 the butter in a skillet over medium heat. Add cucumber, carrots, tomato, and garlic. Cook and stir until vegetables are tender, about 10 minutes. Add rice and ham; cook and stir until heated through and well combined, about 10 minutes. Transfer rice mixture to a bowl.
Beat eggs, salt, and pepper together in a medium bowl. Heat remaining butter in the same skillet over medium heat. Pour in the egg mixture; swirl to coat the bottom of the skillet. Cook until edges are mostly set, about 2 minutes. Lift the omelet with a spatula as the center rises; tilt the skillet so that uncooked eggs run to the center of the pan. Cook until set, 2 to 4 minutes.
Place the rice mixture on top of the omelet. Fold omelet over the rice using a spatula and wrap into a cocoon shape. Place on a plate; sprinkle parsley on top.