Preheat the oven to 450 degrees F. Grease a sheet pan with the olive oil.
Cut the fish fillets into 2 1/2-inch-long and 1-inch-wide strips.
Combine the flour, salt, and pepper in a shallow bowl. Set aside.
Combine the breadcrumbs, parsley and butter with a fork in a separate shallow bowl. Add the Parmesan and stir to combine.
Beat the eggs in a third bowl.
Bread the fish by giving the strips a good coating of the seasoned flour. Tap off any excess. Dip in the beaten eggs until coated. Roll the egg-coated fish in the crumb mixture. If need be, press the breadcrumbs into the fish. Place on the oiled sheet pan.
Bake for 10 minutes, then flip the fish sticks and continue cooking until the breading is deep golden and the fish is cooked through, about another 8 minutes.