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Dissolve 1½ tablespoons salt in 2 quarts cold water in large container. Add beans and soak at room temperature for at least 8 hours or up to 24 hours.
Drain beans in colander and rinse well. Combine beans, 10 cups fresh water, and 1 tablespoon salt in Dutch oven and bring to boil over high heat. Reduce heat to medium and cook at gentle simmer until beans are barely al dente, 40 to 50 minutes. (During simmer, bubbles should just break surface of water.)
Turn off heat, cover pot, and let beans steep until tender, 20 to 30 minutes. Drain beans in colander. (Beans can be refrigerated in airtight container for up to 3 days. Alternatively, beans can be cooled, transferred to zipper-lock bags, and frozen for up to 1 month.)
Remove tuna from container and discard packing oil. Coarsely flake tuna into medium bowl.
Add 3½ cups beans (reserve remainder for another use), oil, parsley, shallot, vinegar, pepper flakes, and remaining ¼ teaspoon salt and stir to combine. Serve.