  We have calculated spin relaxation rates in parabolic quantum dots due to the
phonon modulation of the spin-orbit interaction in presence of an external
magnetic field. Both, deformation potential and piezoelectric electron-phonon
coupling mechanisms are included within the Pavlov-Firsov spin-phonon
Hamiltonian. Our results have demonstrated that, in narrow gap materials, the
electron-phonon deformation potential and piezoelectric coupling give
comparable contributions as spin relaxation processes. For large dots, the
deformation potential interaction becomes dominant. This behavior is not
observed in wide or intermediate gap semiconductors, where the piezoelectric
coupling, in general, governs the spin relaxation processes. We also have
demonstrated that spin relaxation rates are particularly sensitive to the
Land\'e $g$-factor.
