The goal of our work was to study a circularly bent, weakly guiding, multimode optical fiber with a parabolic refractive index profile. With this in mind, the second-order corrections to propagation constants of longitudinally perturbed arbitrary dielectric waveguide’s modes were found using the perturbation theory. Based on that general result, a simple analytic equation describing the corrections to the propagation constants of the modes in the bent parabolic optical fiber was derived. It was shown that the increments of squares of mode propagation constants were the same for all modes. Moreover, the increments of mode propagation constants’ differences in the bent fiber were proportional to those in the straight fiber. The proportionality coefficient was independent of the mode number. The obtained results are of high importance for development of optical fiber sensors, in which fiber bending is possible.