The role of various internet-of-things (IoT) devices responsible for data collection and reporting becomes more important in the era of Industry 4.0. Due to the various advantages (e.g., wide coverage, robust security, etc.), the cellular networks have been continuously evolved to accommodate IoT scenario. In IoT scenario, connection establishment is essential and primary for enabling IoT devices to communicate with centralized unit (e.g., base station (BS)). This connection establishment procedure in cellular networks, random access procedure, is generally operated in a contention-based manner. So, it is vulnerable to simultaneous connection requests from multiple IoT devices to the BS, which becomes worse as the contention participants increase. In this article, we newly propose a resource-efficient parallelized random access (RePRA) procedure for resource-efficiently ensuring reliable connection establishment in cellular-based massive IoT networks. Key features of our proposed technique are twofold: (1) Each IoT device simultaneously performs multiple RA procedures in parallel to improve connection establishment success probability, and (2) the BS handles excessive use of radio resources based on newly proposed two types of redundancy elimination mechanisms. Through extensive simulations, we evaluate the performance of our proposed technique in terms of connection establishment success probability and resource efficiency under various combinations of control parameters. Consequently, we verify the feasibility of our proposed technique for reliably and radio-efficiently supporting a large number of IoT devices.