Scrub the clams under cold running water, discarding any that fail to close to the touch. In a large bowl, combine the clams, coarse salt, and enough water to cover the clams. Let stand for at least 30 minutes or up to two hours so that the clams release any sand trapped in their shells. Drain and set aside the clams.
Then, in a deep skillet, heat the olive oil over high medium-heat. Add the garlic and saute, stirring often, for about one minute, or until golden. Sprinkle in the paprika and breadcrumbs and saute a couple of minutes longer, until ingredients are very fragrant and golden. Add clams and white wine. Cover and let steam for a few minutes, just until the clams open. Discard any clams that don't open.
Check broth for seasoning. Sprinkle with parsley and serve clams with their broth and some crusty sourdough bread on the side. Lemon wedges are optional.