Purpose: To introduce a dual-domain reconstruction network with V-Net and K-Net for accurate MR image reconstruction from undersampled k-space data. Methods: Most state-of-the-art reconstruction methods apply U-Net or cascaded U-Nets in image domain and/or k-space domain. Nevertheless, these methods have following problems: (1) Directly applying U-Net in k-space domain is not optimal for extracting features in k-space domain; (2) Classical image-domain oriented U-Net is heavy-weight and hence is inefficient to be cascaded many times for yielding good reconstruction accuracy; (3) Classical image-domain oriented U-Net does not fully make use information of encoder network for extracting features in decoder network; and (4) Existing methods are ineffective in simultaneously extracting and fusing features in image domain and its dual k-space domain. To tackle these problems, we propose in this paper (1) an image-domain encoder-decoder sub-network called V-Net which is more light-weight for cascading and effective in fully utilizing features in the encoder for decoding, (2) a k-space domain sub-network called K-Net which is more suitable for extracting hierarchical features in k-space domain, and (3) a dual-domain reconstruction network where V-Nets and K-Nets are parallelly and effectively combined and cascaded. Results: Extensive experimental results on the challenging fastMRI dataset demonstrate that the proposed KV-Net can reconstruct high-quality images and outperform current state-of-the-art approaches with fewer parameters. Conclusions: To reconstruct images effectively and efficiently from incomplete k-space data, we have presented a parallel dual-domain KV-Net to combine K-Nets and V-Nets. The KV-Net is more lightweight than state-of-the-art methods but achieves better reconstruction performance.