We present the supernova (SN) sample and Type-Ia SN (SN Ia) rates from the Cluster Lensing And Supernova survey with Hubble (CLASH). Using the Advanced Camera for Surveys and the Wide Field Camera 3 on the Hubble Space Telescope (HST), we have imaged 25 galaxy-cluster fields and parallel fields of non-cluster galaxies. We report a sample of 27 SNe discovered in the parallel fields. Of these SNe, ~13 are classified as SN Ia candidates, including four SN Ia candidates at redshifts z > 1.2. The results are consistent with the rates measured by the HST/GOODS and Subaru Deep Field SN surveys. We model these results together with previous measurements at z < 1 from the literature. We also test DTD models produced by an assortment of published binary population synthesis (BPS) simulations. The shapes of all BPS double-degenerate DTDs are consistent with the volumetric SN Ia measurements, when the DTD models are scaled up by factors of 3-9.