We exploit ionization-parameter mapping (IPM) as a powerful tool to measure the optical depth of star-forming H II regions. Our simulations using the photoionization code Cloudy and our new, SURFBRIGHT surface-brightness simulator demonstrate that this technique can directly diagnose most density-bounded, optically thin nebulae using spatially resolved emission-line data. We apply this method to the Large and Small Magellanic Clouds (LMC and SMC), using the data from the Magellanic Clouds Emission Line Survey. We generate new H II region catalogs based on photoionization criteria set by the observed ionization structure in the [S II]/[O III] ratio and H alpha surface brightness. The luminosity functions from these catalogs generally agree with those from H alpha-only surveys. We then use IPM to crudely classify all the nebulae into optically thick versus optically thin categories, yielding fundamental new insights into Lyman-continuum (LyC) radiation transfer. We find that in both galaxies, the frequency of optically thin objects correlates with H alpha luminosity, and that the numbers of these objects dominate above log L/(erg s(-1)) >= 37.0. The frequencies of optically thin objects are 40% and 33% in the LMC and SMC, respectively. Similarly, the frequency of optically thick regions correlates with H I column density, with optically thin objects dominating at the lowest N(H I). The integrated escape luminosity of ionizing radiation is dominated by the largest regions and corresponds to luminosity-weighted, ionizing escape fractions from the H II region population of >= 0.42 and >= 0.40 in the LMC and SMC, respectively. These values correspond to global galactic escape fractions of 4% and 11%, respectively. This is sufficient to power the ionization rate of the observed diffuse ionized gas in both galaxies. Since our optical depth estimates tend to be underestimates, and also omit the contribution from field stars without nebulae, our results suggest the possibility of significant galactic escape fractions of LyC radiation.