Place each veal cutlet in between 2 pieces of plastic wrap and pound using the spiky side of a meat mallet. This is not to further flatten the meat, but to tenderize it. Lightly sprinkle with 1/2 teaspoon salt and 1/2 teaspoon pepper.
Set up the breading station. Whisk together the flour, 1/4 teaspoon salt and 1/4 teaspoon pepper in a shallow baking dish, and whisk together the mustard and eggs in another shallow baking dish. Combine the breadcrumbs with 1/4 teaspoon salt and 1/4 teaspoon pepper in a third baking dish.
Dredge each veal cutlet first in the flour, then through the egg mixture, and lastly through the breadcrumbs. Transfer onto a baking sheet lined with a cooling rack and refrigerate for 15 minutes.
Preheat the oven to 300 degrees F. Heat 2 tablespoons of the oil over medium-high heat in a large cast-iron skillet. Melt 2 tablespoons of the butter into the oil until it starts to bubble, and then add 1 of the breaded veal cutlets and cook until golden brown on both sides and cooked through, flipping once and moving the pan constantly, 3 to 4 minutes total. Transfer to a baking sheet lined with a cooling rack, season with salt and pepper and keep warm in the oven. Cook the remaining cutlets, making sure to wipe out the pan and use new oil and butter each time. Keep adding the finished veal cutlets to the baking sheet in the oven as you cook them.
To serve, top the wiener schnitzel with the parsley and lemon wedges.