Position a rack in the center of the oven and heat the oven to 200°F. Line a baking sheet with paper towels and set aside.
In a small bowl, stir together the mayonnaise, red onion, relish, parsley, capers, and lemon juice. Cover and refrigerate.
Line up three wide shallow dishes. Put the flour in the first; beat the eggs lightly in the second; put the breadcrumbs in the third. Season the tilapia with salt and pepper. Working with one fillet at a time, coat it with flour and shake off the excess. Then dip the fillet in the egg and dredge it in the breadcrumbs, patting them on to help them adhere. Set each breaded fillet on a plate or tray as you finish it.
Pour the oil 1/4 inch deep into a 10- to 12-inch skillet. Heat over medium heat until it is hot but not smoking, or until a breadcrumb sizzles and browns quickly when dropped in the oil. Cook two of the tilapia fillets until they are crisp and browned, 2 to 3 minutes on each side. Transfer to the baking sheet, sprinkle with salt, and keep warm in the oven while you cook the remaining two fillets.
Serve immediately with lemon wedges and the tartar sauce.