Preheat oven to 400 degrees. Butter an 8-inch square (or other shallow 2-quart) baking dish for extra stuffing.
Heat butter in a large skillet over medium heat; add onions, celery, 1 1/2 teaspoons salt, and 1/4 teaspoon pepper. Cook, tossing often, until beginning to brown, 12 to 15 minutes.
Add mushrooms. Cover; cook until they release liquid, 5 to 7 minutes. Uncover; cook, tossing occasionally, until tender and liquid has evaporated, 10 to 12 minutes more. Transfer to a large bowl; let cool completely.
Add bread, broth, parsley, and 3/4 cup water to bowl; toss well. Season with salt and pepper; mix in eggs.
Place all stuffing in a buttered 9-by-13-inch (or other shallow 4-quart) baking dish.
Put covered dish in oven. Bake 25 minutes. Remove foil; bake until golden, 20 to 25 minutes.