With the rapid growth of the New Space era, the increasing number and diversity of Low Earth Orbit (LEO) satellites demand generic and reusable antenna control software capable of stable and accurate tracking across heterogeneous orbital and antenna environments. Previous works have primarily focused on two-axis antenna systems, relying on step-based tracking or Two-Line Element (TLE)-based orbit prediction. However, such approaches suffer from manual initialization requirements and gimbal lock at high elevation angles, limiting continuous tracking performance. Although three-axis antenna structures have been proposed to address these issues, their lack of software-level generality and experimental validation restricts practical reuse across antenna platforms. This paper presents a generic three-axis parabola antenna control software that integrates TLE-based orbit prediction, a gimbal lock avoidance algorithm, and an antenna specification reflection function. By decoupling antenna-specific parameters from the core tracking logic, the proposed system enables flexible adaptation to diverse antenna configurations. Experimental validation using a real three-axis antenna and simulations demonstrates high tracking accuracy, achieving an average gain error within 0.466 dBm for GEO-KOMPSAT-2A and stable signal strength above 55 dBm for the LEO satellite ARIRANG-5. Furthermore, analysis of 11,469 gimbal-lock-prone satellites confirms robust avoidance performance across diverse orbital conditions.