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, to train these models, one relies on back-propagation through time, which entails unfolding the network over many time steps, making the process of conducting credit assignment considerably more challenging. Furthermore, the nature of back-propagation itself does not permit the use of non-differentiable activation functions and is inherently sequential, making parallelization of the underlying training process very difficult. In this work, we propose the Parallel Temporal Neural Coding Network, a biologically inspired model trained by the local learning algorithm known as Local Representation Alignment, that aims to resolve the difficulties and problems that plague recurrent networks trained by back-propagation through time. Most notably, this architecture requires neither unrolling nor the derivatives of its internal activation functions. We compare our model and learning procedure to other online back-propagation-through-time alternatives (which also tend to be computationally expensive), including real-time recurrent learning, echo state networks, and unbiased online recurrent optimization, and show that it outperforms them on sequence modeling benchmarks such as Bouncing MNIST, a new benchmark we call Bouncing NotMNIST, and Penn Treebank. Notably, our approach can, in some instances, even outperform full back-propagation through time itself as well as variants such as sparse attentive back-tracking. Furthermore, we present promising experimental results that demonstrate our model's ability to conduct zero-shot adaptation.