Warming climate and increasing desertification urge the identification of genes involved in heat and dehydration tolerance to better inform and target biodiversity conservation efforts. Comparisons among extant desert-adapted species can highlight parallel or convergent patterns of genome evolution through the identification of shared signatures of selection. We generate a chromosome-level genome assembly for the canyon mouse (Peromyscus crinitus) and test for a signature of parallel evolution by comparing signatures of selective sweeps across population-level genomic resequencing data from another congeneric desert specialist (Peromyscus eremicus) and a widely distributed habitat generalist (Peromyscus maniculatus), that may be locally adapted to arid conditions. We identify few shared candidate loci involved in desert adaptation and do not find support for a shared pattern of parallel evolution. Instead, we hypothesize divergent molecular mechanisms of desert adaptation among deer mice, potentially tied to species-specific historical demography, which may limit or enhance adaptation. We identify a number of candidate loci experiencing selective sweeps in the P. crinitus genome that are implicated in osmoregulation (Trypsin, Prostasin) and metabolic tuning (Kallikrein, eIF2-alpha kinase GCN2, APPL1/2), which may be important for accommodating hot and dry environmental conditions.