This paper is a continuation of our previous paper. It is concerned with the global existence and the optimal temporal decay estimates for the Cauchy problem of the following multidimensional parabolic conservation laws[formula] Hereu(t,x)=(u1(t,x),…,un(t,x))tis the unknown vector,fj(u)=(fj1(u),…,fjn(u))t(j=1,2,…,N) are arbitraryn×1 smooth vector-valued flux functions defined inBr(u), a closed ball of radiusrcentered at some fixed vectoru∈Rn, andDis a constant, diagonalizable matrix with positive eigenvalues. Our results show that if the flux functionfj(u) satisfiesfj(u)/|u−u|s∈L∞(Br(u),Rn),j=1,2,…,Nfor somes>2+1/N,u∈Rn, then foru0(x)−u∈L∞∩L1(RN,Rn) with ‖u0(x)−u‖L1(RN,Rn)sufficiently small, the above Cauchy problem (*) admits a unique globally smooth solutionu(t,x) andu(t,x) satisfies the following temporal decay estimates. For eachk=0,1,2,… [formula]HereDk=∑|α|=k(∂|α|/∂xα11···∂xαNN). The above decay estimates are optimal in the sense that they coincide with the corresponding decay estimates for the solution to the linear part of the corresponding Cauchy problem.