We consider the problem of exchanging messages in two-way relay (TWR) systems when the sources are correlated binary sequences. In a TWR system, two users communicate simultaneously in both directions to exchange their messages with the help of a relay. Harnessing the fact that the users have access to their own non-compressed messages as side information, each user can compress its message according to the Slepian-Wolf coding strategy by using low-density parity-check codes, particularly, the syndrome approach. Through numerical examples, we show that the proposed scheme offers significant improvements in compression rates compared to the existing solutions in the literature.