In medical imaging, parameterized 3D surface models are of great interest for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. By solving the Yamabe equation with the Ricci flow method, we can conformally parameterize a brain surface via a mapping to a multi-hole disk. The resulting parameterizations do not have any singularities and are intrinsic and stable. To illustrate the technique, we computed parameterizations of cortical surfaces in MRI scans of the brain. We also show the parameterization results are consistent with constraints imposed on the mappings of selected landmark curves, and the resulting surfaces can be matched to each other using constrained harmonic maps. Unlike previous planar conformal parameterization methods, our algorithm does not introduce any singularity points.