Heat oven to 375 degrees. On a rimmed baking sheet, toss bread with 2 tablespoons oil and season with salt and pepper. Bake until golden brown and toasted, 10 to 12 minutes.
Meanwhile, in a large pot, heat remaining 2 tablespoons oil over medium heat. Add shallots, celery, and garlic and cook, stirring, until tender, 6 to 8 minutes. Deglaze with wine, scraping up any brown bits with a wooden spoon.
Add 2 cups water and bring to a boil. Season with salt and pepper. Add mussels and cook, covered, until shells have opened, about 2 minutes. Remove mussels with a slotted spoon and set aside.
Once cool enough to handle, remove mussels from shells and return to broth (discard shells). Stir in parsley and adjust seasoning. In a large bowl, combine bread and broth, toss, and let stand 10 minutes.
Transfer mixture to an oiled 2-quart baking dish and drizzle with oil. Cover with parchment-lined foil and bake 30 minutes; uncover and continue baking until top is golden and stuffing is firm, about 25 minutes more.
Change It Up
Use roasted leeks instead of mussels for a vegetarian riff; swap in Pernod for the wine.