The Team for Advanced Flow Simulation and Modeling (T*AFSM) at Rice University has been using the Stabilized Space-Time Fluid-Structure Interaction (SSTFSI) they developed to model parachute aerodynamics. The complexity of ringsail parachutes requires additional techniques for successful modeling of the reefed stages. Methods developed for this purpose include sequential shape determination, which is an iterative method for determining a shape and corresponding flow field, and coupled FSI using a circumferentially symmetrized traction applied to the parachute. In addition to modeling the reefed stages, these methods provide a suitable starting point for full FSI computations. A multiscale sequentially-coupled FSI computation, together with cable symmetrization, can be used to obtain a refined structural mechanics solution where needed. Furthermore, pressure distribution generation can be used to match structural shapes to drop test observations.