We focus on fluid-structure interaction (FSI) modeling of the ringsail parachutes to be used with the Orion spacecraft. The geometric porosity of the ringsail parachutes with ring gaps and sail slits is one of the major computational challenges involved in FSI modeling. We address the computational challenges with the latest techniques developed by the Team for Advanced Flow Simulation and Modeling (T Black star AFSM) in conjunction with the Stabilized Space-Time Fluid-Structure Interaction (SSTFSI) technique. We investigate the performance of the three possible design configurations of the parachute canopy, carry out parametric studies on using an over-inflation control line (OICL) intended for enhancing the parachute performance, discuss rotational periodicity techniques for improving the geometric-porosity modeling and for computing good starting conditions for parachute clusters, and report results from preliminary FSI computations for parachute clusters. We also present a stability and accuracy analysis for the Deforming-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation, which is the core numerical technology of the SSTFSI technique.