The brain connectivity network can be represented as a graph to reveal its intrinsic topological properties. While classical graph theory provides a powerful framework for examining brain connectivity patterns, it often focuses on low-order graphical indicators and pays less attention to high-order topological metrics, which are crucial to the comprehensive understanding of brain topology. In this paper, we capture high-order topological features via a graphical topology analysis framework for brain connectivity networks derived from functional Magnetic Resonance Imaging (fMRI). Several high-order metrics are examined across varying sparsity levels of binary graphs to trace the evolution of brain networks. Topological phase transitions are primarily investigated that reflect brain criticality, and a novel indicator called "redundant energy" is proposed to measure the chaos level of the brain. Extensive experiments on diverse datasets from healthy controls validate the reproducibility and generalizability of our framework. The results demonstrate that around critical points, classical graph theoretical indicators change sharply, driven by crucial brain regions that have high node curvatures. Further investigations on fMRI of subjects with and without Parkinson's disease uncover significant alterations in high-order topological features which are further associated with the severity of the disease. This study provides a fresh perspective on studying topological architectures of the brain, with the potential to expand our comprehension on brain function in both healthy and diseased states.