Calcium-rich (Ca-rich) transients are a new class of supernovae (SNe) that are known for their comparatively rapid evolution, modest peak luminosities, and strong nebular calcium emission lines. Currently, the progenitor systems of Ca-rich transients remain unknown. Although they exhibit spectroscopic properties not unlike core-collapse Type Ib/c SNe, nearly half are found in the outskirts of their host galaxies, which are predominantly elliptical, suggesting a closer connection to the older stellar populations of SNe Ia. In this paper, we present a compilation of publicly available multiwavelength observations of all known and/or suspected host galaxies of Ca-rich transients ranging from far-UV to IR, and use these data to characterize their stellar populations with prospector. We estimate several galaxy parameters including integrated star formation rate, stellar mass, metallicity, and age. For nine host galaxies, the observations are sensitive enough to obtain nonparametric star formation histories, from which we recover SN rates and estimate probabilities that the Ca-rich transients in each of these host galaxies originated from a core-collapse versus Type Ia-like explosion. Our work supports the notion that the population of Ca-rich transients do not come exclusively from core-collapse explosions, and must either be only from white dwarf stars or a mixed population of white dwarf stars with other channels, potentially including massive star explosions. Additional photometry and explosion site spectroscopy of larger samples of Ca-rich host galaxies will improve these estimates and better constrain the ratio of white dwarf versus massive star progenitors of Ca-rich transients.