Preheat charcoal or gas grill. Chop the garlic and then make a mash in a mortar and pestle with a little butter and olive oil. Add remaining butter, olive oil and parsley. Mix well.
Cut the sourdough bread in thick slices (1" to 1 ¼"). Brush both side and the end of each piece with the garlic butter mixture. Sprinkle generously with finely grated parmesan.
Place on a hot grill for about one minute per side. Don't walk away or it will burn.