<p>By applying a basic splitting to a matrix transformed by a series of Givens rotations, we propose a new parameterized accelerated overrelaxation (PAOR) method for solving large-scale block two-by-two linear systems from time-periodic parabolic optimal control problems. Its convergence is established under reasonable conditions. Furthermore, we propose a class of efficient preconditioners derived from PAOR and analyze the spectral properties of its preconditioned matrices. On the basis of these favorable properties of the preconditioned matrix, we develop a Chebyshev accelerated PAOR method to further improve the efficiency of PAOR. Additionally, its convergence results are also derived based on the error bounds. Several numerical experiments on test problems from time-periodic parabolic problems and other applications are reported, which demonstrates the efficiency of our iterative methods and preconditioners.</p>