  Topological quantum phase transitions (TQPTs) in a material induced by
external perturbations are often characterized by band touching points in the
Brillouin zone. The low-energy excitations near the degenerate band touching
points host different types of fermions while preserving the topological
protection of surface states. An interplay of different tunable topological
phases offers an insight into the evolution of the topological character. In
this paper, we study the occurrence of TQPTs as a function of hydrostatic
pressure in CdGeSb$_2$ and CdSnSb$_2$ chalcopyrites, using the first-principles
calculations. At ambient pressure, both materials are topological insulators
having a finite band gap with inverted order of Sb-$s$ and Sb-$p_x$,$p_y$
orbitals of valence bands at the $\Gamma$ point. On the application of
hydrostatic pressure the band gap reduces, and at the critical point of the
phase transition, these materials turn into Dirac semimetals. On further
increasing the pressure beyond the critical point, the band inversion is
reverted making them trivial insulators. The pressure-induced change in band
topology from non-trivial to trivial phase is also captured by L\"{u}ttinger
model Hamiltonian calculations. Our model demonstrates the critical role played
by a pressure-induced anisotropy in frontier bands in driving the phase
transitions. These theoretical findings of peculiar coexistence of multiple
topological phases in the same material provide a realistic and promising
platform for the experimental realization of the TQPT.
