Heat the oven to 450 degrees. Whisk the egg, flour, salt pepper and allspice in a large bowl until blended. Add the squash and parsley, stirring until thoroughly combined.
Brush a large rimmed baking sheet generously with oil. Spoon 2 tablespoons of the squash mixture per fritter onto the baking sheet, spreading gently. Lightly brush the tops with oil.
Bake for 10 minutes or until lightly browned. Turn and continue baking an additional 3 to 5 minutes or until browned on both sides.