Parabolic pulse propagation in a dispersion-decreasing fiber with normal group velocity dispersion (ND-DDF) has been studied when the third-order dispersion (TOD) is included. The results show that the characteristic of exact self-similarity and linear chirp will be lost due to the third-order dispersion. However, by using the dispersion compensation technique, one can obtain efficient and high-quality compressed pulses. The analytic results have been confirmed using numerical simulations.