Ruddlesden–Popper nickelates have emerged as a promising platform for high-temperature superconductivity, yet the role of lattice degrees of freedom in their correlated normal state remains largely unexplored. Here, we report the observation of a finite phonon thermal Hall effect in the trilayer nickelate La${}_4$Ni${}_3$O${}_{10}$ at ambient pressure. Remarkably, the thermal Hall response is strongly enhanced below the density-wave transition at $T^{*}\approx 140$ K, exhibiting two distinct plateaus in the thermal Hall resistivity, suggesting a strong coupling between density-wave order and lattice excitations. Based on the band crossings between phonons and magnons and the analysis of interlayer Ni-O chains associated with apical oxygen, magnon-phonon hybridization and charge-lattice coupling are proposed as two leading candidate mechanisms, while the specific mechanism remains an open question. These results establish a direct link between density-wave order and phonon transport in nickelates, highlight the active role of lattice degrees of freedom in their correlated normal state, and provide a sensitive probe of intertwined charge, spin, and lattice orders, with implications for understanding both their superconductivity and the multiple possible origins of phonon thermal Hall signals.