In this study, we introduce a novel parallel processing framework for real-time point cloud ground segmentation on FPGA platforms, aimed at adapting LiDAR algorithms to the evolving landscape from mechanical to solid-state LiDAR (SSL) technologies. Focusing on the ground segmentation task, we explore parallel processing techniques on existing approaches and adapt them to real-world SSL data handling. We validated frame-segmentation based parallel processing methods using point-based, voxel-based, and range-image-based ground segmentation approaches on the SemanticKITTI dataset based on mechanical LiDAR. The results revealed the superior performance and robustness of the range-image method, especially in its resilience to slicing. Further, utilizing a custom dataset from our self-built Camera-SSLSS equipment, we examined regular SSL data frames and validated the effectiveness of our parallel approach for SSL sensor. Additionally, our pioneering implementation of range-image ground segmentation on FPGA for SSL sensors demonstrated significant processing speed improvements and resource efficiency, achieving processing rates up to 50.3 times faster than conventional CPU setups. These findings underscore the potential of parallel processing strategies to significantly enhance LiDAR technologies for advanced perception tasks in autonomous systems. Post-publication, both the data and the code will be made available on GitHub.