Bring broth and water to simmer in medium saucepan. Reduce heat to lowest setting and cover to keep warm.
Heat oil in large saucepan over medium heat until shimmering. Add half of grated beets, onion, and ¾ teaspoon salt and cook until vegetables are softened, 5 to 7 minutes. Stir in barley and cook, stirring often, until aromatic, about 4 minutes. Stir in garlic and thyme and cook until fragrant, about 30 seconds. Stir in wine and cook until fully absorbed, about 2 minutes.
Stir in 3 cups warm broth. Simmer, stirring occasionally, until liquid is absorbed and bottom of pan is dry, 22 to 25 minutes. Stir in 2 cups warm broth and simmer, stirring occasionally, until liquid is absorbed and bottom of pan is dry, 15 to 18 minutes.
Add beet greens and continue to cook, stirring often and adding remaining broth as needed to prevent pan bottom from becoming dry, until greens are softened and barley is cooked through but still somewhat firm in center, 5 to 10 minutes. Off heat, stir in remaining grated beets and Parmesan. Season with salt and pepper to taste and sprinkle with parsley. Serve.