We investigate the fibrillization process for amyloid tau fragment peptides (VQIVYK) by applying the discontinuous molecular dynamics method to a system of 48 VQIVYK peptides modeled using a new protein model/force field, PRIME20. The aim of the article is to ascertain which factors are most important in determining whether or not a peptide system forms perfect coherent fibrillar structures. Two different directional criteria are used to determine when a hydrogen bond occurs: the original H-bond constraints and a parallel preference H-bond constraint that imparts a slight bias towards the formation of parallel versus antiparallel strands in a β-sheet. Under the original H-bond constraints, the resulting fibrillar structures contain a mixture of parallel and antiparallel pairs of strands within each β-sheet over the whole fibrillization temperature range. Under the parallel preference H-bond constraints, the β-sheets within the fibrillar structures are more likely to be parallel and indeed become perfectly parallel, consistent with X-ray crystallography, at a high temperature slightly below the fibrillization temperature. The high temperature environment encourages the formation of perfect fibril structures by providing enough time and space for peptides to rearrange during the aggregation process. There are two different kinetic mechanisms, template assembly with monomer addition at high temperature and merging/rearrangement without monomer addition at low temperature, which lead to significant differences in the final fibrillar structure. This suggests that the diverse fibril morphologies generally observed in vitro depend more on environmental conditions than has heretofore been appreciated.