Many temporal logic properties of interest involve both state and action predicates and only hold under suitable fairness assumptions. Temporal logics supporting both state and action predicates such as the Temporal Logic of Rewriting (TLR) can be used to express both the desired properties and the fairness assumptions. However, model checking such properties directly can easily become impossible for two reasons: (i) the exponential blowup in generating the Büchi automaton for the implication formula including the fairness assumptions in its condition easily makes such generation unfeasible; and (ii) often the needed fairness assumptions cannot even be expressed as propositional temporal logic formulas because they are parametric, that is, they correspond to universally quantified temporal logic formulas. Such universal quantification is succinctly captured by the notion of localized fairness; for example, fairness localized to the parameter o in object fairness conditions. We summarize the foundations and present the language design and implementation of the new Maude LTLR Model Checker under localized fairness. This is the first tool we are aware of which can model check temporal logic properties under parametric fairness assumptions.