Real-time time-dependent density functional theory (RT-TDDFT) is known to be hindered by the very small time step (attosecond or smaller) needed in the numerical simulation, which significantly limits its range of applicability in the study of ultrafast dynamics. In this paper, we demonstrate that RT-TDDFT calculations can be significantly accelerated using a combination of two techniques: parallel transport gauge and implicit time integrator, while neither technique would be sufficiently effective on its own. Using absorption spectrum and Ehrenfest dynamics calculations in a planewave basis set for example, we show that the new method can utilize a time step that is on the order of 10-100 attoseconds, and is no less than 5-10 times faster when compared to standard explicit time integrators.