Cortical surface parcellation for fetal brains is essential for the understanding of neurodevelopmental trajectories during gestations with regional analyses of brain structures and functions. This study proposes the attention-gated spherical U-net, a novel deep-learning model designed for automatic cortical surface parcellation of the fetal brain. We trained and validated the model using MRIs from 55 typically developing fetuses [gestational weeks: 32.9 ± 3.3 (mean ± SD), 27.4-38.7]. The proposed model was compared with the surface registration-based method, SPHARM-net, and the original spherical U-net. Our model demonstrated significantly higher accuracy in parcellation performance compared to previous methods, achieving an overall Dice coefficient of 0.899 ± 0.020. It also showed the lowest error in terms of the median boundary distance, 2.47 ± 1.322 (mm), and mean absolute percent error in surface area measurement, 10.40 ± 2.64 (%). In this study, we showed the efficacy of the attention gates in capturing the subtle but important information in fetal cortical surface parcellation. Our precise automatic parcellation model could increase sensitivity in detecting regional cortical anomalies and lead to the potential for early detection of neurodevelopmental disorders in fetuses.