With the emergence of the 5G/6G communications, edge computing has attracted increasing research interests in recent years. To provide pervasive 5G/6G edge computing services, numerous edge servers are required for service coverage, and the deployment cost can be $>$10000 times larger than the deployment cost of the 4 G infrastructure. To address this fundamental limit, we propose ParallEdge, a deployment scheme that employs mobile edge servers for cost-effective service coverage. ParallEdge is designed based on the observation that the processing delay and server moving delay become comparable in many computing-intensive applications in the 5G/6G edge computing. Unlike the traditional “move-then-process” frameworks, ParallEdge follows a “move-while-processing” paradigm, which exploits the parallelism between task processing and server movement to enable resource sharing among more users. Moreover, with the joint optimization of path planning and task scheduling for multiple mobile servers, the deployment cost for service coverage can be further reduced. We analyze the approximation gap of the proposed algorithm, and conduct extensive simulation experiments based on the real-world application data, and the results show that ParallEdge can significantly reduce deployment cost and improve resource utilization compared to the state-of-the-art schemes.