  The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a
Lie bialgebra structure on the underlying loop algebra and its parabolic
subalgebras. In this paper we classify all classical twists of the induced Lie
bialgebra structures in terms of Belavin-Drinfeld quadruples up to a natural
notion of equivalence. To obtain this classification we first show that the
induced bialgebra structures are defined by certain solutions of the classical
Yang-Baxter equation (CYBE) with two parameters. Then, using the
algebro-geometric theory of CYBE, based on torsion free coherent sheaves, we
reduce the problem to the well-known classification of trigonometric solutions
given by Belavin and Drinfeld. The classification of twists in the case of
parabolic subalgebras allows us to answer recently posed open questions
regarding the so-called quasi-trigonometric solutions of CYBE.
