We consider a general system of hyperbolic-parabolic balance laws, which contains both second order dissipation (viscosity) and lower order damping. As dictated by physics, the viscosity matrix and the Jacobian matrix of the lower order term are usually rank deficient. We study the Green's function of its linearization around a constant equilibrium state. The Green's function is constructed using heat kernels along equilibrium characteristics directions and delta functions along frozen characteristics direction. A detailed error estimate is given in space-time pointwise sense. Our result applies to polyatomic gas flows.