I do think you are right to consider parabolic solar cookers for making flat bread. I am sorry to say that I have never eaten Iraqi style flat bread. However, I assume it requires a high temperature to cook properly, and high temperatures are not easy to obtain in many types of solar cookers.
In our way of thinking, there are three important categories of solar cookers--panel cookers (such as the SCI CooKit), box cookers and concentrating (often parabolic or paraboloid) cookers. Panel cookers are not good for the flat breads I know about. Most box cookers do not get hot enough, although I think they could be specially designed for the purpose. Parabolic cookers do seem like the most likely.
You may already know about our Solar Cooking Archive site on the internet. If you click on the section called "build a solar cooker" you will find instructions for building a variety of box cookers, panel cookers, and--at the bottom--some parabolic cookers. The parabolic cookers shown tend to be the low-cost, low-technology type and may not really serve your purposes fully. But studying those designs may give you some ideas.
Another section of the Solar Cooking Archive is called "design and science of solar cookers." In that section, you may be interested in the articles called "Making a Parabolic Reflector out of a Flat Sheet" and "A Parabolic Cooker for Unattended Cooking." If you are interested in the mathematics and theory of parabolic designs, you might like to read the article called "Theories Behind a Chinese Reflective Solar Cooker."
The most common type of parabolic solar cooker in the world is likely to be the "butterfly" type with two not-quite-parabolic reflectors. These are very common in China. You might find something of interest in our latest newsletter which discusses a very popular Chinese solar cooker and includes a photo. You can find that article on the internet [http://solarcooking.org/newsletters/scrnov06.htm#Chinese_company here.