We calculate the 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 (DP) and piezoelectric (PE) electron-phonon couplings are included in the Pavlov-Firsov spin-phonon Hamiltonian. We demonstrate that the spin relaxation rates are particularly sensitive with the Landé g-factor, and that for InAs dots the DP can be necessarily considered in the spin relaxation analysis. Low-temperature (T ~ 0) relaxation rates are found to be small and to depend strongly on size, g-factor, and on magnetic field.
