We study a general system of hyperbolic-parabolic balance laws in m space dimensions (m ≥ 1). The system has rank deficient viscosity matrices and a lower order term whose Jacobian matrix is rank deficient as well. We consider the Cauchy problem when initial data are small perturbations of a constant equilibrium state. Under a set of reasonable assumptions including Kawashima-Shizuta condition, we establish the existence of solution global in time via energy method. The proposed assumptions are sufficiently general for applications to physical models such as electro-magneto flows and physical gas flows. In particular, we study the gas flow with an internal non-equilibrium mode besides the translational non-equilibrium. The general result in this paper recovers the existing results in literature on hyperbolic-parabolic conservation laws and hyperbolic balance laws, respectively, as two special cases.