Spherical harmonics are commonly used in the construction of multi-resolution representations of complex spherical shapes such as brain surface meshes. A key step in generating such representations for a spherical mesh is to construct a one-to-one map onto a sphere. A parametrization inevitably introduces local distortions such as stretching and compression, so that some regions can be severely undersampled or oversampled. As such, parametrizations with minimal metric distortion are desirable because they yield more accurate representations. In this paper, we use a spherical parametrization method that explicitly minimizes the overall distortion penalizing stretching and compression symmetrically. The optimization process uses a hierarchical and iterative scheme. We show numerically that the proposed parametrizations lead to accurate spherical representations of various brain structures.