We present X-ray spectral analyses for three Seyfert 2 active galactic nuclei (AGNs), NGC 424, NGC 1320, and IC 2560, observed by NuSTAR in the 3-79 keV band. The high quality hard X-ray spectra allow detailed modeling of the Compton reflection component for the first time in these sources. Using quasi-simultaneous NuSTAR and Swift/XRT data, as well as archival XMM-Newton data, we find that all three nuclei are obscured by Compton-thick material with column densities in excess of ~5 × 1024 cm-2, and that their X-ray spectra above 3 keV are dominated by reflection of the intrinsic continuum on Compton-thick material. Due to the very high obscuration, absorbed intrinsic continuum components are not formally required by the data in any of the sources. We constrain the intrinsic photon indices and the column density of the reflecting medium through the shape of the reflection spectra. Using archival multi-wavelength data we recover the intrinsic X-ray luminosities consistent with the broadband spectral energy distributions. Our results are consistent with the reflecting medium being an edge-on clumpy torus with a relatively large global covering factor and overall reflection efficiency of the order of 1%. Given the unambiguous confirmation of the Compton-thick nature of the sources, we investigate whether similar sources are likely to be missed by commonly used selection criteria for Compton-thick AGNs, and explore the possibility of finding their high-redshift counterparts.