We report the pressure and temperature dependence of the phonon density-of-states in superconducting Ca_0.6Na_0.4Fe₂As₂ (T_c=21 K) and the parent compound CaFe₂As₂ using inelastic neutron scattering. We observe no significant change in the phonon spectrum for Ca_0.6Na_0.4Fe₂As₂ at 295 K up to pressures of 5 kbar. The phonon spectrum for CaFe₂As₂ shows softening of the low-energy modes by about 1 meV when decreasing the temperature from 300 to 180 K. There is no appreciable change in the phonon density of states across the structural and antiferromagnetic phase transition at 172 K. These results, combined with our earlier temperature dependent phonon density of states measurements for Ca_0.6Na_0.4Fe₂As₂, indicate that the softening of low-energy phonon modes in these compounds may be due to the interaction of phonons with electron or short-range spin fluctuations in the normal state of the superconducting compound as well as in the parent compound. The phonon spectra are analyzed with ab initio and empirical potential calculations giving partial densities of states and dispersion relations.