Future communication networks will require increased flexibility, scalability, and data computation capabilities to adequately respond to the growing number of service demands. Advanced mixed-integer network resource optimization models and algorithms are required to meet these requirements. The purpose of this article is to introduce a hybrid quantum-classical computing framework for addressing future network resource optimization issues. We begin by discussing the fundamentals of quantum computing and its parallelism. Following that, we discuss in detail the hybrid quantum-classical computing paradigm. Then, we discuss the potential applications of the proposed paradigm for network resource optimization, including network function virtualization (NFV), multi-access edge computing/fog/cloud computing, and cloud radio access networks (C-RANs). Finally, we discuss the difficulties associated with the design and implementation of hybrid quantum-classical algorithms.