We investigate the temporal accelerating self-imaging effect for a train of Airy pulses propagating in optical fibers. The group velocity of the Airy pulses is varying during propagation, resulting in a parabolic time-space trajectory of self-imaging. The acceleration is determined by the main-lobe width of the Airy pulses. Meanwhile, the self-imaging distance depends both on the main-lobe width and the time interval of the pulses. In addition, the trajectory of self-imaging can be modified by imposing a linearly varying phase on the input pulse train. By applying third-order dispersion, we also realize the temporally magnified self-imaging of the Airy pulse trains. As the Airy pulses possess infinite energy, the self-imaging can be observed for infinite times. If the pulses are truncated to have finite-energy, the self-imaging maintains only in a limited distance. The study provides a promising way to control the self-imaging of optical pulses and may find applications in optical communication and signal processing systems.