We present a high-order parametric surface finite element method (SFEM) for solving the tangential surface Stokes equations, based on a penalty formulation. Our approach systematically lifts stable finite element pairs from the Euclidean setting to the surface context, enabling a general inf-sup stability analysis that encompasses a broad class of discretizations. We identify how discretization errors are influenced by three key factors: the best-approximation properties of the finite element spaces, the choice of penalty parameters, and the accuracy of the geometric approximation of the surface. Several discretization variants are compared within this framework. In particular, we highlight an isogeometric formulation that requires only geometric information of the discrete surface and achieves optimal-order convergence in tangential norms.