In this paper, a computational model for large amplitude vibrations of a parametrically excited carbon nanotube (CNT) is developed. The continuous model includes geometric and electrostatic nonlinearities. The Galerkin discretization is used to transform the nonlinear partial differential equation to a finite degrees of freedom system which is numerically solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). The influence of higher modes on the nonlinear dynamics of the considered resonator is investigated in order to retain the number of modes which will be used by the HBM+ANM procedure. It is shown that at least two modes are required in order to predict accurately the CNT frequency responses.