What is the best argument for maximum parsimony method in phylogenetic tree construction?
The idea of parsimony is that when constructing phylogenetic trees a simple hypothesis (e.g., four evolutionary changes are necessary to connect two taxa) is more likely to be true than a more complex hypothesis (e.g., 15 evolutionary changes) (source: Berkeley University).

The reason is that for a certain taxon to evolve, there must be a certain number of evolutionary changes. In the end, this happens by more or less stochastic processes (Dingli &amp; Pacheco, 2011) that randomly generate genetic changes (mutations). The characteristics of better fit individuals survive, while those of the least fit are terminated from the gene pool. Hence, every additional change that must be adopted to create a new taxon from another means that the chance that that happens decreases exponentially; it's all a matter of statistics. The less the number of changes needed, the higher the chance that that might have actually happened.      

Reference
- Dingli &amp; Pacheco, BMC Biology (2011);9:41