With a brush, scrub the mussels well and remove their beards. Place the shells in a 4-quart, wide-mouthed pot with the dried kelp, water, and 1 tablespoon of rice wine. Cover the pot and, over high heat, cook the mussels until the shells open (you can hear them do this), which should take about 2 minutes from the time the water boils. Discard any shells that don’t open.
Pour the broth through a cloth- or paper-lined strainer or colander into a 1-quart measuring cup. Add basic sea stock to the broth to make a total of 3 cups of liquid, and season with the soy sauce and 1 tablespoon of rice wine.
Remove the mussels from their steamed open shells, making sure that all traces of beard and grit are removed. If the mussels are larger than 1 inch, cut them in half. Reserve the mussel pieces.
Slit the white end of the leek lengthwise in a deep X and rinse under cold water to remove any sandy soil. Cut across the leek at ¼-inch intervals to chop the vegetable coarsely.
In a 2-quart pot, combine the seasoned broth and leek and cook over medium heat for 2 minutes until the vegetable wilts. Add the mussel pieces to the soup. Dissolve the bean paste in a separate bowl with a bit of broth ladled from the pot, then return all to the pot.
Rinse the flat-leafed parsley under cold water and pat dry with paper towels. Tear off the leaves and chop coarsely. Serve the soup hot, garnishing each bowl with a bit of the chopped parsley.