1. Cook the barley according to package directions until very tender. Drain and chill in the refrigerator at least 3 hours.
2. Put the chilled barley in a food processor and process until smooth. Add the egg yolk, Monterey Jack, ¼ cup of the Parmesan, and 1/3 cup of the flour; process just until mixed. Transfer the mixture to a bowl and stir in the grape leaves, salt, and pepper.
3. Beat the egg whites until stiff, but not dry. Gently fold into the barley mixture along with another 1/3 cup of flour.
4. Line a baking sheet with waxed paper. Place the remaining flour in a shallow dish. Using 2 spoons, shape about 2 heaping tablespoons of the barley dough into a small football shape, drop it into the flour, and coat it lightly. Transfer to the lined baking sheet and repeat the process until all the dough has been used.
5. Bring a large pot of salted water to a boil. Drop 6 to 8 dumplings at a time into the water and cook until they float to the top, about 1 minute. Remove the dumplings from the pot with a slotted spoon and drain on paper towels. When all the dumplings have been cooked, arrange them in a single layer in an ovenproof baking dish. Drizzle with the melted butter and sprinkle with the remaining Parmesan cheese. (The dumplings may be prepared up to this point and refrigerated 1 to 2 days before baking.)
6. Preheat the oven to 375°F.
7. Bake the dumplings until they are puffed and light golden brown, 25 to 30 minutes. Sprinkle with the parsley and serve at once.