Preheat the oven to 325°F with a rack in the middle.
Melt the butter in a 12-inch skillet over medium-high heat.
Add the onions, celery, and apples, along with half the salt and pepper.
Cook, stirring occasionally, until softened and lightly browned in spots, 10 to 15 minutes.
Into a large mixing bowl, place the crumbled cornbread and dried bread cubes.
Top with the apple mixture, parsley, time, sage, and remaining salt and pepper.
Mix gently but thoroughly.
Pour the beaten eggs and the broth over the contents of the bowl, and mix again until the bread has absorbed all the liquid.
Spoon the stuffing into a 9x13x2-inch baking dish.
Bake, uncovered, for 45 minutes, until crisp on top and tender and custardy inside.