Pour 1 inch water into medium saucepan. Bring to a boil over high heat; add parsnips. Cover; boil 10 minutes or until parsnips are fork-tender. Drain. Place in large bowl. Coarsely mash with fork; set aside.
heat 2 tablespoons butter in small skillet over medium-high heat until melted and bubbly. Add onion; cook and stir until transparent. Stir in flour with wire whisk; heat until bubbly and lightly browned. Whisk in milk; heat until thickened. Stir flour mixture into mashed parsnips. Stir in chives; season with salt and pepper to taste.
Form parsnip mixture into 4 patties. Spread breadcrumbs on plate. Dip patties in bread crumbs to coat all sides evenly. Press crumbs firmly into patties. Place on waxed paper and refrigerate 2 hours.
Heat remaining 2 tablespoons butter and oil in 12-inch skillet over medium-high heat until butter is melted and bubbly. Add patties; cook about 5 minutes on each side or until browned. Transfer to warm serving dish. Garnish as desired.