Drones are imperative for the 5G architecture as a mobile source to expand network coverage and support seamless services, particularly through enabling device-to-device (D2D) communication. Such deployment of drones in D2D settings raises various security threats in drone communication. While the existing D2D communication security standard within the 4G cellular architecture may address some of these issues, the standard includes heavy traffic toward the network core servers. If this security standard is to be adopted in the 5G D2D security services with the same traffic load, it may negatively impact the 5G network performance. Therefore, this paper proposes a lightweight proxy signature-based authentication mechanism for a swarm of drones compatible with the 5G D2D standard mechanisms. This paper proposes a distributed delegation-based authentication mechanism to reduce the traffic overhead toward the 5G core network. In this scheme, the legitimate drones are authorized as proxy delegated signers to perform authentication on behalf of the core network. Furthermore, we propose a mechanism to elect and relocate a new leader relay drone from the existing drone swarm. We implemented the proposed authentication algorithm in the 5G D2D-based communication package over NS-3 while performing the computational calculations on a RaspberryPi3 device to mimic the drone calculation process and delays. The performance of the proposed authentication shows a promising reduction in the authentication time and shows lightweight and reliable compatibility.