Use a food processor fitted with the grating disk or a box grater to shred the parsnips and potatoes into two separate bowls. Place the potatoes in a clean kitchen towel and squeeze out as much liquid as possible.
In a large bowl, stir together the potatoes, parsnips, scallions, eggs, flour, parsley, tarragon, salt, and a few pinches of pepper until well blended.
Heat the oven to 200° and line a baking sheet with paper towels. Heat the oil in a large skillet or on a griddle over medium heat. Working in batches, spoon a scant 1 cup of the veggie mixture into the skillet for each pancake. Flatten the pancakes slightly with a spatula (they should be about 1/2 inch thick), and cook them on each side until well browned and crisp, about 2 to 3 minutes per side. If needed, add more oil as you work. Transfer the pancakes to the prepared baking sheet to drain. Sprinkle them with salt and keep them warm in the oven until they're ready to serve.