Based on two-grid discretizations and domain decomposition techniques, this paper presents three parallel finite element algorithms for numerically solving the steady Navier-Stokes equations with damping term. The basic idea of the present algorithms is to first solve a fully nonlinear problem on a coarse grid to get a coarse grid solution, then solve Stokes, Oseen, and Newton linearized residual problems in parallel in overlapping local fine grid subdomains, and finally update the approximate solution in non-overlapping fine grid subdomains. The effectiveness of the proposed algorithms is demonstrated by some numerical examples.