The physics of core-collapse (CC) supernovae (SNe) and how the explosions depend on progenitor properties are central questions in astronomy. For only a handful of SNe, the progenitor star has been identified in pre-explosion images. Supernova remnants (SNRs), which are observed long after the original SN event, provide a unique opportunity to increase the number of progenitor measurements. Here we systematically examine the stellar populations in the vicinities of 23 known SNRs in the Small Magellanic Cloud (SMC) using the star formation history (SFH) maps of Harris & Zaritsky. We combine the results with constraints on the SNR metal abundances and environment from X-ray and optical observations. We find that 22 SNRs in the SMC have local SFHs and properties consistent with a CC explosion, several of which are likely to have been high-mass progenitors. This result supports recent theoretical findings that high-mass progenitors can produce successful explosions. We estimate the mass distribution of the CC progenitors and find that this distribution is similar to a Salpeter IMF (within the uncertainties), while this result is shallower than the mass distribution found in M31 and M33 by Jennings et al. and Díaz-Rodríguez et al. using a similar approach. Additionally, we find that a number of the SMC SNRs exhibit a burst of star formation between 50 and 200 Myr ago. As these sources are likely CC, this signature may be indicative of massive stars undergoing delayed CC as a consequence of binary interaction, rapid rotation, or low metallicity. In addition, the lack of Type Ia SNRs in the SMC is possibly a result of the short visibility times of these sources, as they may fall below the sensitivity limits of current radio observations.