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The coherence of collective modes, such as phonons, and their modulation of the electronic states are long sought in complex systems, which is a cross-cutting issue in photovoltaics and quantum electronics. In photovoltaic cells and lasers based on metal halide perovskites, the presence of polaronic coupling, i.e., photocarriers dressed by the macroscopic motion of charged lattice, assisted by terahertz (THz) longitudinal optical (LO) phonons, has been intensely studied yet still debated. This may be key for explaining the remarkable properties of the perovskite materials, e.g., defect tolerance, long charge lifetimes and diffusion length. Here we use the intense single-cycle THz pulse with the peak electric field up to $E_{THz}=$1000\,kV/cm to drive coherent band-edge oscillations at room temperature in CH$_3$NH$_3$PbI$_3$. We reveal the oscillatory behavior dominantly to a specific quantized lattice vibration mode at $ω_{\mathrm{LO}}\sim$4 THz, being both dipole and momentum forbidden. THz-driven coherent dynamics exhibits distinguishing features: the room temperature coherent oscillations at $ω_{\mathrm{LO}}$ longer than 1 ps in both single crystals and thin films; the {\em mode-selective} modulation of different band edge states assisted by electron-phonon ($e$-$ph$) interaction; {\em dynamic mode splitting} controlled by temperature due to entropy and anharmonicity of organic cations. Our results demonstrate intense THz-driven coherent band-edge modulation as a powerful probe of electron-lattice coupling phenomena and provide compelling implications for polaron correlations in perovskites.