The role of epistasis in driving adaptation has remained an unresolved problem dating back to the Evolutionary Synthesis. In particular, whether epistatic interactions among genes could promote parallel evolution remains unexplored. To address this problem, we employ an Evolve and Resequence (E&R) experiment, using the copepod Eurytemora affinis, to elucidate the evolutionary genomic response to rapid salinity decline. Rapid declines in coastal salinity at high latitudes are a predicted consequence of global climate change. Based on time-resolved pooled whole-genome sequencing, we uncover a remarkably parallel, polygenic response across ten replicate selection lines, with 79.4% of selected alleles shared between lines by the tenth generation of natural selection. Using extensive computer simulations of our experiment conditions, we find that this polygenic parallelism is consistent with positive synergistic epistasis among alleles, far more so than other mechanisms tested. Our study provides experimental and theoretical support for a novel mechanism promoting repeatable polygenic adaptation, a phenomenon that may be common for selection on complex physiological traits.