We elucidate thermal conductivity along the screw dislocation line, which represents a transport direction inaccessible to classical theories. By using equilibrium molecular dynamics simulations, we uncover a Burgers vector dependent thermal conductivity reduction in silicon carbide. The effect is uncorrelated with the classical modeling and originates in the highly deformed core region, which represents a significant source of anharmonic phonon-phonon scattering. High strain reduces the phonon relaxation time, especially in the longitudinal acoustic branches, and creates an effective internal thermal resistance around the dislocation axis. Our results have implications for designing materials useful for high-temperature electronics and thermoelectric applications.