We have witnessed the tremendous momentum of the second spring of parallel computing in recent years. But, we should remember the low points of the field more than 20 years ago and review the lesson that has led to the question at that point whether parallel computing will soon be relegated to the trash heap reserved for promising technologies that never quite make it in an article entitled the death of parallel computing written by the late Ken Kennedy — a prominent leader of parallel computing in the world. Facing the new era of parallel computing, we should le from the robust history of sequential computation in the past 60 years. We should study the foundation established by the model of Turing machine (1936) and its profound impact in this history. To this end, this paper examines the disappointing state of the work in parallel Turing machine models in the past 50 years of parallel computing research. Lacking a solid yet intuitive parallel Turing machine model will continue to be a serious challenge in the future parallel computing. Our paper presents an attempt to address this challenge by presenting a proposal of a parallel Turing machine model. We also discuss why we start our work in this paper from a parallel Turing machine model instead of other choices.