The atmospheric parameters and abundances of Mg, Si, Ca, and Ti have been determined for 20 stars using the Gaia DR2 parallaxes, high-resolution spectra, and modeling of lines without assuming LTE (non-LTE modeling). A sample of stars with homogeneous data on the abundances of α-process elements has thus been increased to 94. It is shown that applying a non-LTE approach and classical ID atmospheric models with spectroscopically determined gravitational accelerations (log g) based on Fe I and Fe II lines yields reliable results. Analysis of the full sample confirms the conclusions of earlier studies indicating excesses of Mg, Si, Ca, and Ti relative to Fe for halo and thick-disk stars, and strong excesses of these elements for thick-disk stars relative to stars with similar metallicities in the thin disk. New results are also obtained. The ratios [Mg/Fe], [Si/Fe], [Ca/Fe], and [Ti/Fe] in the thick disk remain constant and similar to each other at the level ~ 0.3 when [Fe/H] ≲ —0.4, and fall off when the metallicity becomes higher, suggesting the onset of the production of iron in Type la supernovae. Halo stars have the same [α/Fe] values independent of their distance (within ~ 8 kpc of the Sun), providing evidence for a universal character of the evolution of the abundances of α-process elements in different parts of the Galaxy. The excess abundances relative to iron for halo stars are, on average, the same and at the level ~ 0.3 dex for Mg, Si, Ca, and Ti. These data are important for refining nucleosynthesis models. The scatter of [α/Fe] increases for [Fe/H] ≲ —2.6, but the scatter of the ratios between the different α-process elements remains small, possibly indicating incomplete mixing of nucleosynthesis products during the formation of these stars.