Preheat oven to 250°F. Butter a 3-quart baking dish. Place cubed bread on a baking sheet and bake until slightly dry, 10 to 15 minutes.
Raise oven temperature to 350°F. In a large sauté pan over high heat, melt 1 tablespoon of butter. Add the seaweeds, and season with salt and pepper. Cook, tossing occasionally, until it has released the liquid. Scrape into a medium bowl and set aside, reserving the pan.
Return pan to medium heat and melt the remaining 1 tablespoon of butter. Add the thyme and leeks. Cook until very soft, about 5 minutes more. Return the seaweed to the pan, add broth, and bring to a simmer. Season to taste with salt and pepper. In a small bowl, combine the eggs and half and half. Transfer to a large bowl and let it cool slightly. Add cubed bread and egg mixture and toss until the mixture is combined.
Transfer bread mixture to baking dish. Bake in the center of the oven until golden brown and slightly puffed, 35 to 40 minutes. Top with parsley and serve immediately.