The state-of-the-art accelerators for Convolutional Neural Networks (CNNs) typically focus on accelerating only the convolutional layers, but do not prioritize the fully-connected layers much. Hence, they lack a synergistic optimization of the hardware architecture and diverse dataflows for the complete CNN design, which can provide a higher potential for performance/energy efficiency. Towards this, we propose a novel Massively-Parallel Neural Array (MPNA) accelerator that integrates two heterogeneous systolic arrays and respective highly-optimized dataflow patterns to jointly accelerate both the convolutional (CONV) and the fully-connected (FC) layers. Besides fully-exploiting the available off-chip memory bandwidth, these optimized dataflows enable high data-reuse of all the data types (i.e., weights, input and output activations), and thereby enable our MPNA to achieve high energy savings. We synthesized our MPNA architecture using the ASIC design flow for a 28nm technology, and performed functional and timing validation using multiple real-world complex CNNs. MPNA achieves 149.7GOPS/W at 280MHz and consumes 239mW. Experimental results show that our MPNA architecture provides 1.7x overall performance improvement compared to state-of-the-art accelerator, and 51% energy saving compared to the baseline architecture.