In a large mixing bowl, combine warm water and sugar, stirring until dissolved. Sprinkle the active dry yeast over the mixture and let it sit for about 5-10 minutes until frothy. Gradually add the all-purpose flour, salt, and olive oil, mixing until a soft dough forms. Knead the dough on a floured surface for 8-10 minutes until smooth and elastic.

In a small saucepan, melt the butter over low heat. Add the minced garlic and sauté for 1-2 minutes until fragrant but not browned. Remove from heat and stir in the chopped parsley and grated Parmesan cheese until well combined. Allow the filling to cool to room temperature.

Once the dough has risen, roll it out on a floured surface into a large rectangle, about ¼ inch thick. Spread the cooled garlic filling evenly over the dough, leaving a small border around the edges. Starting at one edge, tightly roll the dough into a log shape, then slice it into 1.5-inch pieces.

Place the sliced rolls in a greased baking dish, leaving some space between each. Cover with a clean kitchen towel and let them rise in a warm spot for 30-40 minutes.

Preheat your oven to 375°F (190°C). Once preheated, brush the tops of the rolls with melted butter. Bake for 20-25 minutes, until they're golden brown and sound hollow when tapped.

Remove the rolls from the oven and allow them to cool in the pan for about 10 minutes. Transfer them to a wire rack. Serve warm.