In a bowl, combine contents of roll mix and yeast packet; stir in water, one egg, cheddar cheese and chilies. Turn onto a floured surface; knead dough until smooth and elastic, about 5 minutes. Cover and let rest for 5 minutes.
Divide into thirds. Shape each into a 14-in. rope. Place ropes on a greased baking sheet and braid; pinch ends to seal and tuck under. Cover and let rise in a warm place until doubled, about 30 minutes. Beat remaining egg; brush over dough. Sprinkle with Parmesan cheese. Bake at 375 degrees F for 30 minutes or until golden brown. Remove from pan to a wire rack.