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Layered magnetic materials have enormous potential in enabling the next generation of high-speed nanoscale information storage and spintronic technologies. In order to realize their promise, it is of critical importance that we understand how lattice structure and magnetic order are coupled, particularly on their intrinsic ultrafast time scales. Here, we use ultrafast optical microscopy to study the coherent lattice dynamics of the pseudo-van der Waals ferrimagnet Mn₃Si₂Te₆. Our results reveal strong coherent phonon oscillations under near-infrared pumping that persist from room temperature down to below the magnetic ordering temperature of ~75 K. An analysis of the frequency of these modes as a function of temperature highlights a significant stiffening of the oscillatory frequency across the critical temperature that cannot be explained using a thermal expansion model. Instead, we suggest that this is due to an exchange-mediated contraction of the lattice, highlighting a strong magneto-structural coupling in this material. This is mirrored in the temperature dependence of incoherent dynamics, measured across the magnetic phase transition. Our findings showcase the ultrafast dynamics in this semiconducting layered ferrimagnet and suggest a new route for their optical control through their intertwined electronic, lattice, and spin degrees of freedom.