Energy internet (EI) is a complex coupled multienergy system; it is essential to investigate its multienergy dispatching optimization issues. To this end, this paper first proposes a novel conception of smart dispatching for EI with a complex cyber-physical-social system (CPSS) network from the perspective of parallel dispatch, called parallel dispatching robot (PDR), and investigates the implementations of PDR based on smart artificial society (SAS) modeling. First, we introduce EI and describe the dispatching issues of EI. Second, we discuss several important concepts supporting the parallel dispatch conception of EI, including knowledge automation (KA), CPSS, and parallel machine learning (PML). On the basis of these, we elaborate the concept of parallel dispatch. Moreover, we construct a large closed-loop feedback control framework of parallel dispatch for EI integrating a CPSS network based on KA and PML. Third, we establish an experimental platform for PDR research based on the proposed parallel dispatch framework. Fourth, we develop the PML-based SAS models of a single PDR in centralized dispatching modes and group PDRs in decentralized dispatching modes to achieve crowd wisdom emergence and performance improvement in current cyber-physical system frameworks of EI. Moreover, we design an external global closed loop for PDR to evaluate its operation stability. Lastly, we conduct a detailed discussion on PDR and offer some prospects for its engineering implementations. The biggest innovation of this paper lies in systematically proposing the smart dispatching concept and framework for complex CPSS-based EI from the perspective of parallel dispatch and thoroughly investigating how to use SAS modeling to implement parallel dispatching and control for EI considering human and social factors, which is a major extension and theoretical improvement to existing single smart wide area robot concept and a preliminary attempt in investigating a shift from Energy 4.0 to Energy 5.0 in China.