In practical engineering, system reliability analysis is highly concerned since many structures or products have multiple failure modes. Accordingly, this paper develops an innovative method for system reliability analysis by parallel learning of influential component limit-state functions with filtered sample region (PLIC-FSR-SYS) based on Kriging modeling. Different from the traditional adaptive learning methods that train only one component in each iteration when constructing the surrogate of the composite limit-state function, a new strategy is explored to adaptively identify several important components in one iteration so as to train them simultaneously. In the meanwhile, a filtering formula is explored to determine the fatal region so that the unimportant samples can be removed to further accelerate the training process. Based on the join forces of parallel learning of influential components and avoiding the training at unimportant samples, PLIC-FSR-SYS can achieve a fairly efficient system reliability analysis with multiple failure modes. Finally, four different case studies, including an engineering application to the ultra-voltage on-load tap-changer, are conducted to prove the effectiveness of the proposed method. The results indicate that compared to traditional adaptive learning methods, the proposed method makes a significant efficiency improvement for system reliability analysis with multiple failure modes.