With the advent of the Internet-of-Things (IoT) era, the ever-increasing number of devices and emerging applications have triggered the need for ubiquitous connectivity and more efficient computing paradigms. These stringent demands have posed significant challenges to the current wireless networks and their computing architectures. In this article, we propose a high-altitude platform (HAP) network-enabled edge computing paradigm to tackle the key issues of massive IoT connectivity. Specifically, we first provide a comprehensive overview of the recent advances in non-terrestrial network-based edge computing architectures. Then, the limitations of the existing solutions are further summarized from the perspectives of the network architecture, random access procedure, and multiple access techniques. To overcome the limitations, we propose a HAP-enabled aerial cell-free massive multiple-input multiple-output network to realize the edge computing paradigm, where multiple HAPs cooperate via the edge servers to serve IoT devices. For the case of a massive number of devices, we further adopt a grant-free massive access scheme to guarantee low-latency and high-efficiency massive IoT connectivity to the network. Besides, a case study is provided to demonstrate the effectiveness of the proposed solution. Finally, to shed light on the future research directions of HAP network-enabled edge computing paradigms, the key challenges and open issues are discussed.