Inter-robot collisions pose a significant safety risk when multiple robotic arms operate in close proximity. We present an online collision avoidance methodology leveraging High-Order Control Barrier Functions (HOCBFs) constructed for safe interactions among 3D convex shapes to address this issue. While prior works focused on using Control Barrier Functions (CBFs) for human-robotic arm and single-arm collision avoidance, we explore the problem of collision avoidance between multiple robotic arms operating in a shared space. In our methodology, we utilize the proposed HOCBFs as centralized and decentralized safety filters. These safety filters are compatible with many nominal controllers and ensure safety without significantly restricting the robots' workspace. A key challenge in implementing these filters is the computational overhead caused by the large number of safety constraints and the computation of a Hessian matrix per constraint. We address this challenge by employing numerical differentiation methods to approximate computationally intensive terms. The effectiveness of our method is demonstrated through extensive simulation studies and real-world experiments with Franka Research 3 robotic arms. The project video is available at this link.