Geometric flows have recently attracted lots of attention from scientific computing communities. One of the most popular schemes for solving geometric flows is the so-called BGN scheme, which was proposed by Barrett, Garcke, and Nurnberg (J. Comput. Phys., 222 (2007), pp. 441--467). However, the BGN scheme only can attain first-order accuracy in time, and how to design a temporal high-order numerical scheme is challenging. Recently, based on a novel approach, we have successfully proposed temporal high-order, BGN-based parametric finite element method for solving geometric flows of curves/surfaces. Furthermore, we point out that the shape metrics (i.e., manifold distance), instead of the function norms, should be used to measure numerical errors of the proposed schemes. Finally, ample numerical experiments demonstrate that the proposed BGN-based schemes are high-order in time in terms of the shape metric, and much more efficient than the classical BGN schemes.