Preheat oven to 325 degrees Fahrenheit.
In a frying pan, heat oil over medium heat. Add onions and sauté until golden.
In a bowl, combine sautéed onions with potatoes, parsnip, and carrot.
Add eggs, salt, and pepper, and mix to combine.
Working with one capon at a time, place face down. Top with a heaping tablespoon of filling (or a bit more, depending on the size of the chicken).
Roll up tightly and place in a baking pan. Brush with oil. Repeat with remaining capons and filling.
Cover pan tightly and bake three hours.