What are the major weaknesses of the parsimony and distance matrix methods of phylogeny ...

The main problem with parsimony is that, by definition, it assumes the most parsimonious (shortest or simplest) evolutionary pathway, while it is almost certainly true that this is not invariably the pathway taken. Still, as a guiding principle, parsimony has much to recommend it. Distance methods make no such assumptions about how evolution should work, but on the other hand assume that simple unweighted similarity and dissimilarity reflect evolutionary relatedness and that evolutionary rate is constant across lineages. We know that this is not true, owing to selective constraints, convergence, etc.