A technique for center manifold reduction of nonlinear delay differential equations with time-periodic coefficients is presented. The DDEs considered here have at most cubic nonlinearities multiplied by a perturbation parameter. The periodic terms and matrices are not assumed to have predetermined norm bounds, thus making the method applicable to systems with strong parametric excitation. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The accuracy of the method is demonstrated with a nonlinear delayed Mathieu equation, a milling model, and a single inverted pendulum with a periodic retarded follower force and nonlinear restoring force in which the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.