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We present a detailed density functional theory based calculations of the structural, elastic, lattice dynamical, thermophysical, and optoelectronic properties of ternary semiconductors CaZn$_2$X$_2$ (X = N, P, As) in this paper. The obtained lattice parameters are in excellent agreement with the experimental values and other theoretical findings. These elastic constants satisfy the mechanical stability criteria. Moreover, many thermophysical parameters of these compounds are estimated, including the Debye temperature, average sound velocity, melting temperature, heat capacity, lattice thermal conductivity, etc. The comprehensive analysis of the elastic constants and moduli show that CaZn$_2$X$_2$ compounds possess reasonably good machinability, relatively high Vickers hardness and relatively low Debye temperature. The phonon dispersion curves and phonon density of states are investigated for the first time for the compounds CaZn$_2$P$_2$ and CaZn$_2$As$_2$. It is observed from the phonon dispersion curves that the bulk CaZn$_2$X$_2$ (X = N, P, As) compounds are dynamically stable. Electronic properties have been studied through the band structures and electronic energy density of states. The electronic band structures show that CaZn$_2$N$_2$ and CaZn$_2$As$_2$ possess direct band gaps while the compound CaZn$_2$P$_2$ show indirect band gap. The bonding characters of CaZn$_2$X$_2$ (X = N, P, As) compounds are investigated. Energy dependent optical parameters exhibit good correspondence with the electronic energy density of states features. We have thoroughly discussed the reflectivity, absorption coefficient, refractive index, dielectric function, optical conductivity and loss function of these semiconductors. The optical absorption, reflectivity spectra and the refractive index of CaZn$_2$X$_2$ (X = N, P, As) show that the compounds hold promise to be used in optoelectronic devices.