We investigate the dynamics of Wilson-Cowan networks with distributed delays, initially focusing on networks with two nodes before extending our analysis to networks with an arbitrary number of nodes. Our primary aim is to understand the combined effects of distributed delays and network connectivity on the network's dynamic behavior. Our model describes the interactions of excitatory and inhibitory populations across nodes, with connection strengths represented by specific weight matrices. To account for the distributed nature of delays within the network, we incorporate a delay kernel that integrates past states over time. We first assume identical delay distributions across all connections, allowing us to analyze the model without imposing specific network architectures. This simplification serves as a starting point for understanding network dynamics before addressing the more complex case of delay heterogeneity. Through stability and bifurcation analyses, we reveal how distributed delays and network connectivity jointly influence the network's dynamic behavior. As an application, we model abnormal neural oscillations associated with Parkinson's disease, highlighting the fundamental roles of delays and connectivity patterns in understanding the mechanisms underlying pathological brain rhythms.