Remove outer leaves; cut dark-green ends from leeks. Trim roots, keeping leaves attached. Halve lengthwise; rinse to remove grit.
Melt butter in a large skillet over medium heat. Add leeks to pan and arrange cut side down in a single layer. Cook until softened and light golden, 3 to 4 minutes. Turn leeks and cook on the other side until softened, 3 minutes. Season with salt and pepper. Turn the leeks again so they are cut side down and peel off any papery outer layers. Add stock and wine, stirring to deglaze pan. Cook for 1 minute. Bring to a boil, reduce to a simmer, and cover with a parchment paper round. Cook until the leeks are tender when pierced with a knife, about 15 minutes.
Remove parchment, raise heat, and cook until liquid is reduced, about 1 to 2 minutes. Sprinkle with coarse salt and parsley; serve.