This paper considers two parallel supply chains with interacting demand streams. Each supply chain consists of one supplier and one retailer. The two demand streams are jointly described with a vector autoregressive time-series process in which they interact and their respective innovation errors correlate contemporaneously. For each supply chain, we develop insights into when and how much the supplier and the retailer can improve on their forecasting accuracy if the external demand history of the other supply chain is utilized. When this external demand history is not available or made available after a time lag, we develop a partial process and a delayed process to characterize the demand structure that the retailer can recover from the available demand histories. Our results show that the external demand history of the other supply chain always helps the retailer make better forecasts when demand streams interact; however, the enhanced information alters the retailer's order process, which may produce larger forecasting errors for the supplier. Conditions are established for the supplier to benefit from the external demand history of the other supply chain.