We introduce Sparse Physics Informed Backpropagation (SPInProp), a new class of methods for accelerating backpropagation for a specialized neural network architecture called Low Rank Neural Representation (LRNR). The approach exploits the low rank structure within LRNR and constructs a reduced neural network approximation that is much smaller in size. We call the smaller network FastLRNR. We show that backpropagation of FastLRNR can be substituted for that of LRNR, enabling a significant reduction in complexity. We apply SPInProp to a physics informed neural networks framework and demonstrate how the solution of parametrized partial differential equations is accelerated.