Distributed detection with dependent observations is always a challenging problem. The problem of detection with shared information has many applications when sensors have overlapped measurements, e.g., when distributed detection is performed in a security system where sensors have overlapped coverages. For this shared information scenario, we investigate the distributed detection problem in parallel fusion networks. The design problem is how to best utilize the common information at both the local sensors and the fusion center to achieve best possible performance. We derive the necessary condition for the optimal sensor decision rules for all sensors. In addition, we investigate the system performance by comparing the optimal rules with suboptimal rules for distributed detection of a constant signal corrupted by Gaussian noise. The numerical results obtained by conducted examples con- firm the optimality of the derived decision rules.