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Purpose: To introduce a novel deep learning based approach for fast and high-quality dynamic multi-coil MR reconstruction by learning a complementary time-frequency domain network that exploits spatio-temporal correlations simultaneously from complementary domains. Theory and Methods: Dynamic parallel MR image reconstruction is formulated as a multi-variable minimisation problem, where the data is regularised in combined temporal Fourier and spatial (x-f) domain as well as in spatio-temporal image (x-t) domain. An iterative algorithm based on variable splitting technique is derived, which alternates among signal de-aliasing steps in x-f and x-t spaces, a closed-form point-wise data consistency step and a weighted coupling step. The iterative model is embedded into a deep recurrent neural network which learns to recover the image via exploiting spatio-temporal redundancies in complementary domains. Results: Experiments were performed on two datasets of highly undersampled multi-coil short-axis cardiac cine MRI scans. Results demonstrate that our proposed method outperforms the current state-of-the-art approaches both quantitatively and qualitatively. The proposed model can also generalise well to data acquired from a different scanner and data with pathologies that were not seen in the training set. Conclusion: The work shows the benefit of reconstructing dynamic parallel MRI in complementary time-frequency domains with deep neural networks. The method can effectively and robustly reconstruct high-quality images from highly undersampled dynamic multi-coil data ($16 \times$ and $24 \times$ yielding 15s and 10s scan times respectively) with fast reconstruction speed (2.8s). This could potentially facilitate achieving fast single-breath-hold clinical 2D cardiac cine imaging.