We introduce the Dynamic Spectral-Parsing Graph Neural Network (DspGNN), a novel model that incorporates spectral-designed graph convolution for representation learning and edge regression on Discrete Time Dynamic Graphs (DTDGs). Our first major contribution is the optimization of spectral-designed methods to better capture evolving spectral information on DTDGs. Secondly, to solve the computational challenge of performing eigendecomposition on large DTDGs, we propose a novel technique, Active Node Mapping, that proves to be both simple and effective. Our model consistently outperforms baseline methods on three publicly available datasets for edge regression tasks.