In a large mixing bowl, combine the warm water (about 110°F to 115°F) with the sugar and active dry yeast. Stir and let sit for 5 to 7 minutes until foamy.
Add the all-purpose flour and salt to the yeast mixture. Pour in 1 tablespoon melted unsalted butter. Stir until a rough dough forms. Add more flour if too sticky.
Turn dough onto a floured surface. Knead for 8 to 10 minutes until smooth, elastic, and slightly tacky but not sticky.
Lightly oil a mixing bowl, place dough inside, cover with towel or plastic wrap. Let rise in a warm place until doubled, about 1 to 1.5 hours.
Melt remaining 1 tablespoon butter in a small saucepan over low heat. Add minced garlic and cook gently for 1 to 2 minutes until fragrant. Remove from heat, stir in chopped parsley, and cool slightly.
Punch down risen dough and turn out on floured surface. Roll into 12x8 inch rectangle. Slice into 12 equal squares.
Brush each square with garlic butter mixture. Sprinkle grated Parmesan cheese over each piece. Stack squares, cut into three columns of four squares each. Place stacks upright in baking pan.
Cover pan and let rolls rise again for 30 to 40 minutes until puffy.
Preheat oven to 375°F (190°C). Bake rolls for 20 to 25 minutes until golden brown and cooked through. Brush tops with remaining garlic butter after baking.
Let rolls cool slightly before serving. Enjoy warm by pulling apart pieces to savor garlicky, buttery flavor.
