Abstract: Understanding the coupling between spin and phonons is critical for controlling the lattice thermal conductivity (kappa ( l )) in magnetic materials, as we demonstrate here for CrX3 (X = Br and I) monolayers. We show that these compounds exhibit large spin-phonon coupling (SPC), dominated by out-of-plane vibrations of Cr atoms, resulting in significantly different phonon dispersions in ferromagnetic (FM) and paramagnetic (PM) phases. Lattice thermal conductivity calculations provide additional evidence for strong SPC, where particularly large kappa ( l ) is found for the FM phase. Most strikingly, PM and FM phases exhibit radically different behavior with tensile strain, where kappa ( l ) increases with strain for the PM phase, and strongly decreases for the FM phase-as we explain through analysis of phonon lifetimes and scattering rates. Taken all together, we uncover the high significance of SPC on the phonon transport in CrX3 monolayers, a result extendable to other 2D magnetic materials, that will be useful in further design of thermal spin devices.