This paper deals with the solution of fluid-structure interaction problems using an algorithm consisting in the coupling between two solvers, each one related to the fluid and the structure respectively. Due to the difficult decomposition of the domain into sub-domains, we consider for each environment a parallel multi-splitting algorithm which corresponds to an unified presentation of sub-domain methods with or without overlapping. Such method combines several contractant fixed point mappings and, we show that, under appropriate assumptions, each fixed point mapping are contractive in finite dimensional spaces normed by Hilbertian and non-Hilbertian norms. Moreover, we show in a novel way that such study is valid for the parallel synchronous and more generally asynchronous large scale linear systems arising in the solution of fluid-structure interaction problems and can be extended to the case where the displacement of the structure is submitted to some constraints. We perform parallel simulations for a fluid-structure test case on different clusters, considering blocking and non-blocking communications. The performances of the parallel simulations are presented and analyzed.