Theoretical study of structural stability and elastic properties of α- and β-MgPd3 intermetallic compounds as well as their hydrides have been carried out based on density functional theory. The results indicate α-MgPd3 is more stable than β phase with increased stability in their hydrides. The calculated elastic constants of α-MgPd3 are overall larger than β phase. After hydrogenation, the elastic constants are enlarged. And the elastic moduli exhibit similar tendency. The anisotropy of α-MgPd3 is larger than β phase, and the hydrides demonstrate larger anisotropy. Their ductility follows the order of α-MgPd3H0.5 < α-MgPd3 < β-MgPd3H < β-MgPd3. Compared with β phase, higher Debye temperature of α-MgPd3 implies stronger covalent interaction, and the Debye temperature of hydrides increases slightly. The electronic structures demonstrate that the Pd–Pd interaction is stronger than Pd–Mg, and Pd–H bonds play a significant role in the phase stability and elastic properties of hydrides.