Temporal models based on recurrent neural networks have proven to be quite powerful in a wide variety of applications, including language modeling and speech processing. However, training these models often relies on backpropagation through time (BPTT), which entails unfolding the network over many time steps, making the process of conducting credit assignment considerably more challenging. Furthermore, the nature of backpropagation itself does not permit the use of nondifferentiable activation functions and is inherently sequential, making parallelization of the underlying training process difficult. Here, we propose the parallel temporal neural coding network (P-TNCN), a biologically inspired model trained by the learning algorithm we call local representation alignment. It aims to resolve the difficulties and problems that plague recurrent networks trained by BPTT. The architecture requires neither unrolling in time nor the derivatives of its internal activation functions. We compare our model and learning procedure with other BPTT alternatives (which also tend to be computationally expensive), including real-time recurrent learning, echo state networks, and unbiased online recurrent optimization. We show that it outperforms these on-sequence modeling benchmarks such as Bouncing MNIST, a new benchmark we denote as Bouncing NotMNIST, and Penn Treebank. Notably, our approach can, in some instances, outperform full BPTT as well as variants such as sparse attentive backtracking. Significantly, the hidden unit correction phase of P-TNCN allows it to adapt to new data sets even if its synaptic weights are held fixed (zero-shot adaptation) and facilitates retention of prior generative knowledge when faced with a task sequence. We present results that show the P-TNCN's ability to conduct zero-shot adaptation and online continual sequence modeling.