The coupling between magnons and phonons and the associated phenomena have long been a focus of research in condensed matter physics. Contrary to its recognized role in magnon relaxation, its impact on phonon transport remains largely unexplored. Here, we fill this gap by investigating the effect of magnon-phonon coupling on phonon excitation, relaxation, and transport with magneto-optical reflectometry. Through simultaneous measurements of magnon and phonon populations in magnetic insulator thulium iron garnet, we observe the excitation of excessive phonons driven by non-equilibrium magnons, demonstrating the magnetic control of phonons. Furthermore, our time-resolved experiments reveal the magnetic field-dependent phononic thermal conductivity, signaling the potential of magnetic manipulation of heat transport. Our finding indicates that phonons can be controlled by magnetic means through magnon-phonon coupling and thereby opens a new avenue to harness magneto-thermoelectric effects in magnetic insulators.