In a small bowl, combine the sugar and yeast. Pour the warm water over the mixture and let it stand for 5 to 10 minutes, until the surface is foamy and the yeast smells fragrant.
Meanwhile, in a large bowl or the bowl of a stand mixer, attach the dough hook and combine the flour and salt.
Add the yeast mixture and oil to the center of the dry ingredients. Mix with a wooden spoon or on low speed just until the dough begins to come together.
Transfer the dough to a well-floured surface and knead 10 to 15 minutes, adding flour 1 tablespoon at a time as needed, until the dough is smooth, soft, elastic, and only slightly tacky; or knead in the stand mixer on medium speed 5 to 7 minutes, adjusting with small additions of flour until the dough reaches the same smooth, supple texture.
Place the dough in a lightly greased bowl, cover with plastic wrap, and let rise in a warm spot for about 1 hour, or until doubled in size.
In a small bowl, combine the softened butter, parsley, garlic, and a pinch of salt. Set aside.
Gently punch down to deflate the dough. Divide it into 2 equal portions, place them on a lightly floured surface, and cover with a clean kitchen towel. Let rest for 5 minutes to relax the gluten.
Working with one portion at a time, roll the dough into a 14×8-inch rectangle.
Spread 2 tablespoons of the garlic-butter mixture evenly over the surface. Sprinkle ½ cup Cheddar cheese, 2 tablespoons of Parmesan, and a light dusting of black pepper.
Starting from a long edge, roll the dough into a tight 14-inch log and pinch the seam to seal.
Place the log seam-side down. Using a sharp knife, slice the log into 1-inch sections, cutting almost all the way through but keeping the rolls connected at the base. Grease two 8-inch round cake pans with cooking spray.
Carefully lift the log and transfer it to one of the prepared pans, arranging it into a ring with the cut sides facing upward. Repeat with the remaining dough so each pan holds one ring.
Spread or brush the remaining garlic-butter mixture over the tops of the rolls.
Cover the pans and let the dough rest 15 to 20 minutes while the oven finishes heating.
Meanwhile, heat the oven to 400°F. Bake for 20 to 25 minutes, until the tops are deeply golden and the cheese is bubbling at the edges.
Let cool briefly before serving.