In Python there are very easy to use and efficient libraries for surface parametrization (for instance igl). I wonder if in Mathematica we have a ready-made implementation.

The parametrization is typically given as a UV list $\{\{u_1,v_1\},\{u_2,v_2\},\cdots\}$.

Some sample data for a particular parametrization I am working with is here, with the surface points, the triangle list and the parametrization: