An approach for generating high-resolution a priori maximum parsimony Y-chromosome (chrY) phylogenies based on SNP and small INDEL variant data from massively-parallel short-read (next-generation) sequencing data is described; the tree-generation methodology produces annotations localizing mutations to individual branches of the tree, along with indications of mutation placement uncertainty in cases for which "no-calls" (through lack of mapped reads or otherwise) at particular sites precludes precise phylogenetic placement of mutations. The approach leverages careful variant site filtering and a novel iterative reweighting procedure to generate high-accuracy trees while considering variants in regions of chrY that had previously been excluded from analyses based on short-read sequencing data. It is argued that the proposed approach is also superior to previous region-based filtering approaches in that it adapts to the quality of the underlying data and will automatically allow the scope of sites considered to expand as the underlying data quality improves (e.g. through longer read lengths). Key related issues, including calling of genotypes for the hemizygous chrY, reliability of variant results, read mismappings and "heterozygous" genotype calls, and the mutational stability of different variants are discussed and taken into account. The methodology is demonstrated through application to a dataset consisting of 1292 male samples from diverse populations and haplogroups, with the majority coming from low-coverage sequencing by the 1000 Genomes Project. Application of the tree-generation approach to these data produces a tree involving over 120,000 chrY variant sites (about 45,000 sites if singletons are excluded). The utility of this approach in refining the Y-chromosome phylogenetic tree is demonstrated by examining results for several haplogroups. The results indicate a number of new branches on the Y-chromosome phylogenetic tree, many of them subdividing known branches, but also including some that inform the presence of additional levels along the trunk of the tree. Finally, opportunities for extensions of this phylogenetic analysis approach to other types of genetic data are noted.