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. For more information, see Yancheng Sangli Solar Energy.
The Ethiopians and Eritreans use a clay griddle for preparing their ingera. This type of griddle is not conductive enough to allow it to be heated hot enough and evenly enough by most solar cookers. We have received two reports of the use of cast iron griddles made in China. These worked much better than the ones made of clay. When ingera is cooked in the traditional way over a wood fire, the correct temperature of the griddle is 180°C.