Bring a large pot of water to a boil over high heat, then add 2 generous pinches of salt. Add onions and cook until soft when pierced, 15 - 20 minutes. Drain onions in a colander and set aside to cool enough to handle. Peel onions by trimming root ends and slipping them out of their skins, then transfer to a large baking dish, spreading onions out in a single even layer.
Preheat broiler. Put sugar and 2/3 cup of water into a medium heavy bottomed saucepan, cover pot and boil, swirling pan over high heat several times until sugar dissolves, about 5 minutes. Add butter and 1 teaspoon salt and heat, swirling pan occasionally until butter melts, about 5 minutes more.
Pour sugar butter mixture over onions and broil, basting and turning onions frequently until onions brown and syrup turns pale golden, 20 - 30 minutes. Transfer onions and syrup to a warm serving dish and garnish with parsley sprigs.