Propagation of a modulationally amplified Gaussian pulse is considered inside a birefringent optical fiber. We show analytically that, in addition to a reduction in the parametric gain and a change in the pulse's group velocity, the third-order dispersion allows the generation of a temporal Airy-shaped pulse. Furthermore, pulse spreading in the temporal domain and chirping in the spectral domain both can be controlled through an interplay between third-order dispersion and modulational instability.