In the three-dimensional Dirac semimetal Sr_1-yMn_1-zSb₂, Dirac fermions derived from the Sb plane coexist with breaking of time-reversal symmetry by the Mn plane. A Peierls distortion causes the Sb square-net to form zig-zag chains. Quantum oscillations show the persistence of Dirac fermions, despite the distortion, while calculations for stoichiometric SrMnSb₂ predict that the distortion should gap the Dirac cones. We have previously shown that photoexcitation of SrMnSb₂ with a short optical pulse launches the coherent oscillation of several phonon modes. This suggests that, by using phonons to coherently control the atoms’ positions, one could open or close a gap at a Dirac point on a sub-picosecond timescale. Here we combine first-principles calculations of the electronic structure and of the phonon spectrum with time-resolved and Raman spectroscopy to identify the atomic displacements of several of the coherently-controlled phonon modes. We show that some of these modes oscillate in a way that periodically strengthens and relaxes the Peierls distortion. We calculate the effect of large-amplitude motions in these modes on the electronic structure. For one particular A_g mode, a large-amplitude motion is predicted to close the gap by restoring the Dirac cone near the Y-point.