5G is the beginning of a new era in cellular communication, bringing up a highly connected network with the incorporation of the Internet of Things (IoT). To flexibly operate all the IoT devices over a cellular network, Device-to-Device (D2D) communication standard was developed. However, IoT devices such as drones utilizing 5G D2D services could be a perfect target for malicious attacks as they pose several safety threats if they are compromised. Furthermore, there will be heavy traffic with an increased number of IoT devices connected to the 5G core. Therefore, we propose a lightweight, fast, and reliable authentication mechanism compatible with the 5G D2D ProSe standard mechanisms. Specifically, we propose a distributed authentication with a delegation-based scheme instead of the repeated access to the 5G core network key management functions. Hence, a legitimate drone is authorized by the core network via offering a proxy signature to authenticate itself to other drones. We implemented the proposed protocol in ns-3 that supports 5G D2D-based communication. We also conducted computational calculations on the RaspberryPi3 IoT device to mimic the drone calculation process and delays. The results demonstrate that the proposed protocol is lightweight and reliable.