Preheat the oven to 400 degrees F. Cut off the top third of the head of garlic, exposing the tops of the garlic cloves. Put the garlic on a piece of heavy-duty aluminum foil and drizzle with 1 tablespoon of the oil. Sprinkle with 1/4 teaspoon each salt and pepper and add the thyme sprigs. Wrap tightly with foil and roast until the cloves are golden brown and very soft, about 45 minutes.
Unwrap the garlic and let stand until cool enough to handle. Pour any garlic-flavored oil from the foil pouch into a small bowl; squeeze the roasted cloves into the same bowl, discarding the papery shells. Add the remaining 1 tablespoon oil, the chopped thyme and an additional 1/4 teaspoon each salt and pepper and mash with a fork until smooth.
Lay the bottom half of the bread on a piece of aluminum foil that is long enough to wrap the bread. Spread on half of the roasted garlic mixture and top with half of the Parmesan. Repeat with the remaining bread, garlic mixture and Parmesan. Press the halves together so the fillings are on the inside and wrap with the foil.
Bake until the bread is toasted and the cheese is melted, about 15 minutes. Separate the halves, sprinkle with chopped parsley and cut into 6 pieces.