As a follow-up to the discovery of a new family of Fe-based superconductors, namely, the RE₄Fe₂As₂Te_1−xO₄ (42214) (RE = Pr, Sm, and Gd), we present a detailed ab initio study of these compounds highlighting the role of rare-earth (RE) atoms, external pressure, and Te content on their physical properties. Modifications of the structural, magnetic, and electronic properties of the pure (e.g., x = 0.0) 42214 compounds and their possible correlations with the observed superconducting properties are calculated and discussed. The careful analysis of the results obtained shows that (i) changing the RE atoms allows one to tune the internal pressure acting on the As height with respect to the Fe planes; (ii) similarly to other Fe pnictides, the 42214 pure compounds show an antiferromagnetic-stripe magnetic ground state phase joined by an orthorhombic distortion (not experimentally found yet); (iii) smaller RE atoms increase the magnetic instability of the compounds possibly favoring the onset of the superconducting state; (iv) external pressure induces the vanishing of the magnetic order with a transition to the tetragonal phase and can be a possible experimental route towards higher superconducting critical temperature (T_c); and (v) Te vacancies act on the structural parameters, changing the As height and affecting the stability of the magnetic phase.