México was dominated by the Partido de la Revolución Institucional from 1929 until 2000, when they finally lost their grip on the presidency. The roots of the PRI itself were planted at the end of the Revolution in 1920. While there were certainly leftist groups, and even communist parties , the domination of the PRI prevented them from gaining any serious traction. The better question is why was the PRI so popular for so long? The simple answer is that, after a century of doddling along after independence, the PRI implemented reforms that dramatically increased living standards. Cardenas, for example, gave land back to the people who worked it (in the form of ejidos) and nationalized the oil industry, while subsequent governments instituted Import Substitution Industrialization (ISI), which grew the economy very quickly. Indeed, 1930 to 1970 is known as the Mexican Miracle.

The 1960s brought increasing unrest given that the PRI had ruled for so long, particularly amongst student groups (who had their own interests), culminating in the Tlatelolco massacre of 1968. They would hold power for another 32 years, but this was really the first big Crack, the second being the 1985 earthquake and the disastrous response to it. The collapse of oil prices in the 1980s, the increasing ineffficiencies inherent to ISI, and the implementation of NAFTA (which began the dismantling of the ejido system) only made things worse. There were some marxist rebel groups (the Zapatistas and the Ejército Revolucionario Popular) who gathered some support, particularly in the 1990s, but aside from the Zapatista Revolt in Chiapas in 1994, they were never really a threat.

The most far left political party now is the PRD which has never won the presidency but seems to have a consistent 30% support. It would be difficult to describe them as Marxist, although both it and the PRI belong to the Socialist International.

Lots of great texts on 20th C Mexico - I like The Course of Mexican History by Ruiz.