Preheat oven to 475F (250C), with racks in top and bottom third. Line two baking sheets with aluminum foil; set aside.
Place egg in a wide shallow bowl; season with salt and pepper. In another bowl, combine panko, Old Bay Seasoning, and oil. Dip tilapia into egg, shaking off excess, then into panko mixture, pressing to adhere. Place on prepared baking sheets.
Bake until lightly browned, 12 to 15 minutes, rotating sheets from top to bottom halfway through. Meanwhile, in a small bowl, stir together mayonnaise, parsley, mustard, lemon juice, and horseradish; season with salt and pepper.
Serve fish sticks with sauce and lemon wedges on the side.