In the bowl of a stand mixer fitted with the dough hook, stir together warm water, yeast, and sugar. Let stand until creamy, about 5 minutes.
To the yeast mixture, add 2 Tablespoon softened butter, flour, and salt. Mix on low until the ingredients begin to come together.
Increase the speed to medium and knead until the dough is smooth and elastic, about 5 minutes. (You will know it is ready when the dough starts to make a slapping sound as it hits the sides of the bowl.)
Remove the bowl from the mixer and cover it with a dish towel. Let the dough rise in a warm place until doubled in volume, about 20 to 30 minutes (if the kitchen is on the cooler side, use the higher end of the rise time).
While dough rises, combine 6 Tablespoons butter and garlic in a small saucepan over low-medium heat. Heat until butter is melted and garlic is fragrant.
Brush about half the garlic butter (reserve the second half) over the bottom and sides of 2 cake pans (alternatively, you can bake all of the rolls together in a 9x13 baking dish).
After the dough has completed its first rise, turn it out onto a lightly floured surface and form it into an even ball.
Slice the dough into 16 even portions and gently form them into balls. Place the dough balls in prepared cake pans.
Preheat oven to 400°F / 200°C. Place cake pans near the warm oven and let dough rise for 15 minutes more.
Bake rolls until just starting to turn golden brown on top, 15 minutes.
Remove rolls from the oven and brush generously with remaining garlic butter. Sprinkle with parmesan cheese.
Return rolls to oven and continue baking just until cooked through and deep golden brown, 3 to 5 minutes more.
Sprinkle with chopped parsley if you’d like.
Let rolls cool in the pan for 10 minutes before turning out of the pan. Best served the same day they’re baked.