Cut a slit in each chestnut with scissors or a paring knife. Cook chestnuts in a pot of boiling water for 20 minutes, then drain in a colander. When cool enough to handle, peel off and discard shells and inner brown skins. Quarter each.
Preheat oven to 375 degrees. Crumble cornbread into a large bowl. Melt butter in a large skillet over medium heat. Cook chestnuts, stirring occasionally, until they begin to brown, about 8 minutes. With a slotted spoon, transfer chestnuts to bowl, reserving butter in pan.
Add onions, celery, and shallots to pan; season with salt. Cook, stirring occasionally, until they begin to soften, about 5 minutes. Add apples and continue to cook, stirring occasionally, until apples are soft, about 5 minutes more. Transfer to bowl with cornbread mixture. Add parsley, sage, and enough stock to moisten mixture. Toss to combine and season with salt and pepper.
Transfer stuffing to a buttered 3-quart shallow baking dish. (Stuffing can be prepared to this point up to 1 day ahead; cover with plastic wrap and refrigerate. Return to room temperature and uncover before baking.) Bake until heated through and top is lightly browned, about 30 minutes. Serve hot.