The fluid-structure interaction (FSI) problem has received great attention in the last few years, mainly because it is present in many physical systems, industrial applications, and almost every biological system. In the parallel computational field, outstanding advances have been achieved for the individual components of the problem, allowing, for instance, simulations around complex geometries at very high Reynolds numbers or simulations of the contraction of a beating heart. However, it is not an easy task to combine the advances of both fields, given that they have followed development paths in a rather independent way, and also because physical and numerical instabilities arise when dealing with two highly nonlinear partial differential equations. Nonetheless, in the last few years great advances in the coupled FSI field have been achieved, recognizing the most challenging problems to tackle and enabling a new generation of numerical simulations in aerodynamics, biological systems, and complex industrial devices. Keeping in mind that efficient parallel codes for the individual components already exist, this paper presents a framework to build a massively parallel FSI solver in a multicode coupling partitioned approach, with strong focus in the parallel implementation aspects and the parallel performance of the resulting application. The problem is casted in an algebraic form, and the main points of interest are the parallel environment needed to be able to transfer data among the codes, the location of the exchange surface, and the exchange of information among the parallel applications. The proposed framework has been implemented in the HPC multiphysics code Alya, and the multicode coupling is carried out running separated instances of this code. Two coupling algorithms with different acceleration schemes are revised, and three representative cases of different areas of interest showing the reach of the proposed framework are solved. Good agreement with literature and experiments is obtained. In addition to the numerical validation of the FSI solver, an assessment of the parallel performance of the proposed multicode strategy is done. In particular, a special distribution of the fluid code and solid code MPI processes on the supercomputer nodes based on computing cores overloading is investigated. Finally, a strong scalability test, running up to a 30 million elements case using 1280 MPI processes, is done.