Using a food processor, shred 1 pound peeled parsnips (about 4 cups). In a large bowl, toss together the shredded parsnips, 1 cup thinly sliced scallions, 1/4 cup flour and 3/4 teaspoon each salt and pepper; stir in 2 beaten eggs. In a large, heavy skillet, heat 2 tablespoons vegetable oil over medium-high heat. Gently fold 1/2 cup panko breadcrumbs into the parsnip mixture until just combined. Working in batches and adding more oil as needed, drop rounded tablespoonfuls of the mixture into the skillet 1 inch apart. Flatten slightly with a spatula and fry, turning once, until crisp and golden, about 5 minutes. Serve immediately.