Heat oven to 400° F.
Heat the oil in a large skillet over medium heat. Add the onions and cook until golden brown, about 8 minutes. Add the carrots and celery and cook until soft, about 6 minutes. Add 1 1/2 teaspoons salt, 3/4 teaspoon pepper, the parsley, and thyme and cook for 1 minute.
In a large bowl, combine the bread, vegetables, and broth. Transfer to a baking dish. Bake for 20 minutes, until heated through.
Make-Ahead Note: Prepare the stuffing for baking up to 1 day in advance.