Based on two-grid discretization and domain decomposition, three finite element parallel algorithms for unsteady Navier-Stokes equations are proposed. The key idea of the algorithms is to solve nonlinear problem firstly by Oseen iteration method on a coarse grid, and then to solve Oseen, Newton or Stokes problem in parallel on a fine grid to correct the coarse grid solution at each time step, respectively. Conforming finite element pairs are used for spatial discretization and backward Euler scheme for temporal discretization. Numerical results are shown to verify effectiveness of the algorithms.