In this paper we discuss the problem of computing eigenvectors for matrices in real Schur form using parallel computing. We develop a new parallel algorithm and report on the performance our our MPI based implementation. We have im- plemented a new parallel algorithm for scaling during the backsubstitution phase. We have increased the arithmetic intensity by interleaved the compution of several eigenvectors and by merging the backward substitution and the back-transformation of the eigenvectors.