Heat 2 tablespoons butter with the vegetable oil in a large cast-iron skillet over medium-high heat until almost smoking. Add the onion, season with salt and pepper and cook until translucent, about 4 minutes. Add half of the vegetables and cook, stirring occasionally, until golden, about 5 minutes. Add the corned beef and cook until slightly brown, about 2 minutes.
Meanwhile, mash the remaining vegetables in a bowl with the parsley and 3/4 cup cooking liquid. Add to the skillet and stir to make a large pancake, adding up to 1/2 cup more cooking liquid, if needed. Cook until the hash is dark on the bottom, about 2 minutes.
Flip and continue cooking, turning occasionally and adding more butter if needed, until the hash is crunchy on the outside but still moist in the middle, 5 to 7 minutes. Add 1/4 cup cooking liquid 1 minute before serving.