Zhu proved a duality theorem between level one affine Demazure modules and function rings of torus fixed point subschemes of affine Schubert varieties in affine Grassmannian. Using his methods and results, we prove a similar duality theorem between level one twisted affine Demazure modules and twisted affine Schubert varieties for absolutely special parahoric group schemes. As a consequence, we determine the smooth locus of all twisted affine Schubert varieties for many types of parahoric group scheme. This confirms a conjecture of Haines and Richarz for these types of group schemes. If time permits, I will also talk about how this duality theorem is related to the Frenkel-Kac isomorphism for twisted affine Lie algebras, and also the fusion product for twisted affine Demazure modules. This is a joint work with Marc Besson.