Dissolve 2 tablespoons salt in 3 quarts cold water in large bowl or container. Add beans and soak at room temperature, 8 to 24 hours. Drain and rinse well.
Adjust oven rack to lower-middle position and heat oven to 300 degrees. Using kitchen twine, tie together celery, bay leaf, and thyme. Place sausage and salt pork in medium saucepan and add cold water to cover by 1 inch; bring to boil over high heat. Reduce heat to simmer and cook 5 minutes. Transfer sausages to cutting board, allow to cool slightly, then cut into 1-inch pieces. Remove salt pork from water; set aside.
Heat 2 tablespoons oil in large Dutch oven over medium-high heat until beginning to smoke. Add sausage pieces and brown on all sides, 8 to 12 minutes total. Transfer to medium bowl. Add pork shoulder and brown on all sides, 8 to 12 minutes total. Add onion and carrots; cook, stirring constantly, until onion is translucent, about 2 minutes. Add garlic and tomato paste and cook, stirring constantly, until fragrant, 30 seconds. Return sausage to Dutch oven; add white wine, using wooden spoon to scrape browned bits from bottom of pan. Cook until slightly reduced, about 30 seconds. Stir in tomatoes, celery bundle, and reserved salt pork.
Stir in broth and beans, pressing beans into even layer. If any beans are completely exposed, add up to 1 cup water to submerge (beans may still break surface of liquid). Increase heat to high and bring to simmer. Cover pot, transfer to oven, and cook until beans are tender, about 1½ hours. Remove celery bundle and salt pork and discard. (Alternatively, dice salt pork and return to casserole.) Using large spoon, skim fat from surface and discard. Season with salt and pepper. Increase oven temperature to 350 degrees and bake, uncovered, 20 minutes.
Meanwhile, pulse bread and remaining 2 tablespoons oil in food processor until crumbs are no larger than 1/8 inch, 8 to ten 1-second pulses. Transfer to medium bowl, add parsley, and toss to combine. Season with salt and pepper.
Sprinkle ½ cup bread-crumb mixture evenly over casserole; bake, covered, 15 minutes. Remove lid and bake 15 minutes longer. Sprinkle remaining bread-crumb mixture over top of casserole and bake until topping is golden brown, about 30 minutes. Let rest 15 minutes before serving.