We demonstrate the role of longitudinal optical (LO)-phonon assisted polariton relaxation in reducing the stimulation threshold in strongly coupled microcavities. When the energy of the relaxation bottleneck is one LO-phonon above the ground polariton state, we observe a tenfold improvement of the polariton relaxation rate in the linear regime, and a twofold reduction in the threshold to the nonlinear polariton lasing regime.