In a mixing bowl, combine the chopped corned beef, drained sauerkraut, shredded Swiss cheese, softened cream cheese, and Dijon mustard. Stir until all ingredients are well incorporated. Cover the mixture and refrigerate for about 30 minutes to firm it up.

After chilling, scoop out tablespoon-sized portions of the mixture and roll them into 1-inch balls using your hands. Place each ball on a lined baking sheet, ensuring they are not touching each other.

Prepare your breading station by setting out three shallow bowls. Fill one with flour, another with beaten eggs, and the third with breadcrumbs.

Take each ball and roll it in flour, shaking off any excess. Next, dip it into the beaten eggs for an even coating, and finally roll it in the breadcrumbs until fully covered. Place the coated balls on a tray to rest.

In a deep skillet or heavy-bottomed pot, pour in enough vegetable oil to submerge the balls, about 2-3 inches deep. Heat the oil to 350°F (175°C) over medium heat.

Carefully drop a few coated Reuben Balls into the hot oil, making sure not to overcrowd the pan. Fry them for about 2-3 minutes, turning occasionally, until golden brown and crispy.

While the Reuben Balls are frying, whisk together mayonnaise, ketchup, pickle relish, hot sauce, and smoked paprika in a medium bowl. Taste and adjust seasoning.

Transfer the crispy Reuben Balls to a serving platter, arranging them neatly. Serve warm alongside the spicy Thousand Island dipping sauce, and if desired, garnish with chopped parsley.