In a large mixing bowl combine flour, red pepper, and salt. Using a pastry blender, cut in cheddar cheese and butter until pieces are the size of small peas. Sprinkle 1 tablespoon of the water over part of the mixture. Gently toss with a fork. Push to side of bowl. Repeat until all of the flour mixture is moistened. Shape dough into a ball.
On a lightly floured surface flatten dough with your hands. Roll out dough from center to edges forming a 10-inch square. Brush with egg; sprinkle with Parmesan cheese. Cut dough into 5x1/2-inch strips. Place the strips 1/2 inch apart on a lightly greased baking sheet.
Bake in a 400 degree F oven for 10 to 12 minutes or until golden brown. Cool crackers on wire rack. Makes 40 crackers.