We report the observation of magnetophonon resonances on the background of the longitudinal magneto-conductance in short-period GaAs/AlAs semiconductor superlattices. Both the GaAs and the AlAs longitudinal optical phonons are observed. We show how the enhancement of the magnetophonon effect with electric field is connected to the shift of the electron distribution towards the high-velocity and low-density-of-states region at the midpoint of the reduced Brillouin zone. The observed temperature dependence can be explained by considering the competition between the phonon population and the electron lifetime. The two superposed series of the GaAs and the AlAs optical phonons are shifted when hydrostatic pressure is applied and the relative strength of each series changes as the Gamma miniband comes closer to the X states.