We study a domain decomposition approach for a coupled fluid-structure interaction (FSI) system involving a three-dimensional (3D) Stokes flow and a two-dimensional (2D) fourth-order Euler–Bernoulli (or Kirchhoff) plate. The interaction between the parabolic and hyperbolic partial differential equations occurs at the boundary interface, where coupling conditions enforce the matching of fluid and plate velocities and the Dirichlet trace of the fluid pressure. The main contribution is a domain decomposition (DD) framework for the FSI system in which the plate equation is reformulated as two coupled second-order PDEs. This avoids the use of H2-conforming elements and provides flexibility in the choice of discrete approximation spaces. We analyze well-posedness and temporal stability and present numerical results demonstrating the effectiveness of the proposed method.