AT HOME (UP TO 1 DAY AHEAD)
Preheat oven to 375°. Toss squash with 1 tbsp. oil. Spread out on a baking sheet and roast until tender when pierced with a fork, about 15 minutes.
Meanwhile, heat remaining 2 tbsp. oil in a large frying pan. Cook onion, garlic, mushrooms, and celery, stirring often, until softened and starting to brown. Sprinkle with salt, then transfer to a large mixing bowl.
Add bread cubes, roasted squash, broth, and parsley; toss gently to combine. Transfer to a greased 9- by 13-in. baking dish.
AT THE PARTY
Bake at 350° until warm in the center and top is just toasted and crusty, about 1 hour.
*If you like a heartier flavor, use dark rye. Otherwise, use light rye, a mix of rye and sourdough, or all sourdough.