The spin current in a parabolically confined semiconductor heterojunction quantum wire with Dresselhaus spinorbit coupling is theoretically studied by using the perturbation method. The formulae of the elements for linear and angular spin current densities are derived by using the recent definition for spin current based on spin continuity equation. It is found that the spin current in this Dresselhaus spin-orbit coupling quantum wire is antisymmetrical,which is different from that in R ashba model due to the difference in symmetry between these two models. Some numerical examples for the result are also demonstrated and discussed.