There is a rapid adoption of offshore floating solar as a renewable energy source in the world. An accurate analysis of the hydro-elastic response of these systems is necessary to design them. These floating platforms are assumed to be thin, flexible membranes, allowing lower spatial dimensional solid modelling. Mixed-dimensional fluid-structure interaction models are developed and used to simulate this phenomenon. In a monolithic linear system, the fluid and solid subproblems are solved together at the same time. These systems are ill-conditioned, making the use of the monolithic set of equations difficult to solve using iterative solvers, leading to numerical instability in the solution. The primary source of the numerical instability observed in this coupled system is the added mass effect, where the fluid contributes significant inertia loading to the floating structure. The floating structure's low density and thin geometry exacerbate this effect. A block preconditioning strategy that leverages the fluid-solid interface conditions for the monolithic linearized system is shown, which mitigates this instability and enables efficient solutions for large-scale problems on high-performance computing clusters. The linearized problem is constructed using primitive variables, making it straightforward to extend this approach to non-linear problems. We demonstrate the practical implementation of this framework using Gridap.jl and FROSch. Test cases demonstrating the robustness and parallel scalability of this framework up to 512 cores are presented.