Based on fully overlapping domain decomposition, a parallel finite element algorithm for the unsteady Oseen equations is proposed and analyzed. In this algorithm, each processor independently computes a finite element approximate solution in its own subdomain by using a locally refined multiscale mesh at each time step, where conforming finite element pairs are used for the spatial discretizations and backward Euler scheme is used for the temporal discretizations, respectively. Each subproblem is defined in the entire domain with vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for and hence can be solved in parallel with other subproblems using an existing sequential solver without extensive recoding. The algorithm is easy to implement and has low communication cost. Error bounds of the parallel finite element approximate solutions are estimated. Numerical experiments are also given to demonstrate the effectiveness of the algorithm.