New magnitude bounds of the frequency response functions for the Nonlinear AutoRegressive model with eXogenous input (NARX) are investigated by exploiting the symmetry of the nth-order generalized frequency response function (GFRF) in its n frequency variables. The new magnitude bound of the nth-order symmetric GFRF is frequency-dependent, and is a polynomial function of the magnitude of the first order GFRF. The coefficients of this polynomial function are functions of model parameters. Based on this result, the system output spectrum can also be bounded by an analytical polynomial function of the magnitude of the first order GFRF. The conservatism in the bound evaluations is reduced compared with previous results. Several examples and necessary discussions illustrate the potential application and effectiveness of the new results.