Was the political system of the Mongolian Empire parlamentary and, if so, how did it become parlamentary?
The governmental structure of the Mongolian Empire under Genghis was indeed parliamentary, though non-democratic in that the member chiefs were appointed based on extreme loyalty or bravery. It was called the Kurultai and consisted of the various Mongol chiefs who met with the Kahn to discuss common issues, both foreign and domestic. There were sort of "national" Kurulati, which elected the "Great Kahn," and regional Kurultai, which elected regional leaders. The amount of autonomy these bodies possessed is debatable and dependent upon the situation the Khans found themselves in.
Unlike a modern parliament, the Kurultai also functioned as a war council in which goals and strategies would be discussed. Thus, its main job was to select new leaders and strategize. Less time was spent in lawmaking than in a modern parliament.
The early mongols were very clannish and lived over expanses of land. They squabbled and fought in the same way the nobility of Europe did. The parliamentary system that developed was a way for these tribes to maintain some unity of thought and action. The power of such a body fluctuates greatly depending on the attitudes of those in it and the strength of the individual clans. When united, such as during the later time of Genghis, it could function fairly well. Under the rule of weaker leaders it could lead to fragmentation.