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. We measure volumetric SN Ia rates to redshift 1.8 and add the first upper limit on the SN Ia rate in the range 1.8 < z < 2.4. 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. The best-fitting SN Ia delay-time distribution (DTD; the distribution of times that elapse between a short burst of star formation and subsequent SN Ia explosions) is a power law with an index of -1.00 +0.06(0.09) -0.06(0.10) (statistical) +0.12 -0.08 (systematic), where the statistical uncertainty is a result of the 68% and 95% (in parentheses) statistical uncertainties reported for the various SN Ia rates (from this work and from the literature), and the systematic uncertainty reflects the range of possible cosmic star-formation histories. 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. In contrast, all BPS single-degenerate DTDs are ruled out by the measurements at a >99% significance level.