<p>Iteration mappings provide a unifying language for analyzing how numerical minimization methods cluster toward candidate minimizers, and basin geometry is increasingly used to compare robustness and reliability across methods. Recent work developed strong basins, boundary notions induced by pre-distance functions, and inverse-iteration representations of tolerance basins for exact iteration mappings, including single-valued and multiset-valued cases. Motivated by the practical reality that implementations are <i>inexact</i> (finite precision, approximate derivatives, truncated subproblem solves, nonunique internal decisions, and stochastic perturbations), we develop a robust and parametric theory of basins for inexact iteration mappings modeled as set-valued perturbations. We introduce robust tolerance basins defined by eventual capture into a prescribed neighborhood <i>uniformly over all admissible errors</i>. Under a local contraction hypothesis with contraction factor strictly less than one, we derive explicit disturbance-to-neighborhood bounds showing that every inexact orbit starting sufficiently close to a strong attractor remains in a forward-invariant neighborhood and stays within a computable asymptotic radius determined by the error level and the contraction strength; the bound is sharp in general. We also treat parametric families of iteration mappings and establish Lipschitz-type sensitivity of fixed points and certified robust basin neighborhoods with respect to parameter changes. Finally, we develop a robust inverse-iteration viewpoint: robust basin membership implies a necessary weak-preimage inclusion, while a strong-preimage construction yields a necessary-and-sufficient characterization consistent with the universal quantifier inherent in robustness. Examples and computational recipes, including boundary localization via amplification diagnostics, illustrate how to certify robust interiors and numerically detect thin basin boundaries.</p>