Place parsnips in a large pot, and fill with enough water to cover. Bring to a boil, and cook until parsnips are tender, about 10 minutes. Drain parsnips and mash with a potato masher. Set aside.
Melt butter in a saucepan over medium heat. Add onions; cook and stir until onions are translucent, about 5 minutes. Whisk in the flour, and stir until the mixture becomes paste-like, about 5 minutes. Gradually whisk the milk into the flour mixture, and bring to a simmer over medium heat. Cook and stir until the mixture forms a thick sauce, about 10 minutes.
Combine the sauce, the mashed parsnips, and the bread crumbs. Season the mixture to taste with salt and pepper; form mixture into patties.
Heat oil in a large, deep skillet to 350 degrees F (175 degrees C). Fry patties, turning once, until browned on both sides, about 5 minutes per side. Drain on paper towels; serve hot.