We report GHz longitudinal as well as shear acoustic phonons photoexcitation and photodetection using femtosecond laser pulses in a spin-crossover molecular crystal. From our experimental observation of time domain Brillouin scattering triggered by the photoexcitation of acoustic waves across the low-spin (LS) to high-spin (HS) thermal crossover, we reveal a link between molecular spin state and photoexcitation of coherent GHz acoustic phonons. In particular, we experimentally evidence an unconventional non-thermal pathway for the laser excitation of GHz phonons. We also provide experimental insight on the optical and mechanical parameters evolution across the LS/HS spin crossover temperature T$_{1/2}$.