The excitation spectrum around the pump-only stationary state of a polariton optical parametric oscillator in semiconductor microcavities is investigated by time-resolved photoluminescence. The response to a weak pulsed perturbation in the vicinity of the idler mode is directly related to the lifetime of the elementary excitations. A dramatic increase of the lifetime is observed for a pump intensity approaching and exceeding the optical parametric oscillator threshold. The observations can be explained in terms of a critical slowing down of the dynamics upon approaching the threshold and the following appearance of a soft Goldstone mode in the spectrum.