The growing prominence of electric vehicle (EV) aggregators in the modern power system is drawing more attention towards modeling their behavior in the short-term electricity markets. The demand-side flexibility offered by the EVs can be leveraged to reduce their charging costs. In this paper, the participation of an EV aggregator in the intraday and balancing market is modeled as a multistage stochastic programming problem. The computational complexity introduced by the peculiarities of the intraday market is solved by a  progressive hedging algorithm (PHA), a scenario-based decomposition technique. A randomized scenario sampling approach is implemented to accelerate the PHA which is further improved with a parallel randomized PHA. Finally, an asynchronous version of the parallel randomized PHA is leveraged to speed up the multistage model of EV aggregator trading. We compare the computation time of the modified versions of the PHA algorithm with the conventional PHA for the proposed EV aggregator model. Furthermore, we also show the value of EV aggregator trading in the intraday and balancing markets by comparing its cost to baseline models.