1. Melt vegetable shortening and the butter or margarine in a large, heavy kettle.
2. Stir in potatoes, onion, celery, salt and pepper; cover loosely.
3. Cook slowly, stirring often, 30 minutes, till potatoes are tender.
4. Pour milk over bread crumbs in a small bowl; stir into potato mixture, then stir in eggs. Continue cooking, stirring constantly, until eggs are cooked. (Flecks of cooked egg will show in the stuffing.) Remove from heat.
5. Stir in the parsley. Cool slightly.