This paper presents a novel approach of direction-of-arrival (DoA) estimation for the electronically steerable parasitic array radiator (ESPAR) antennas, using only a single radio-frequency (RF) chain. Starting from the problem formulation in the Bayesian compressive sensing (BCS) framework, the CS measurements are projected onto the beamspace of the unique configuration of the ESPAR antenna. In this work, measurements collected at multiple snapshots are considered. First, we propose to solve the sparse recovery problem by the multi-task BCS . Then, the DoAs are estimated by employing a noise filter on the recovered sparse signal. In this method, the number of sources need not be known a priori, and computation complexity is reduced by avoiding computing the correlation matrix of measurements unlike the traditional DoA estimation techniques. Simulations show that the proposed method can recover closely spaced sources using a small number of noisy snapshots, and it performs better with more sources than other state-of-the-art algorithms.