The Restricted Santa Claus Problem [1] is a classic \(\textsf {NP}\) -hard fair allocation problem, where the goal is to assign a set of indivisible gifts to children in a way that maximizes the minimum total happiness across all children. Each gift provides zero happiness value to some children and a fixed positive happiness value to the remaining children. In this paper, we study the parameterized complexity of the problem with respect to the minimum happiness threshold  \(\tau \) and some structural parameters associated with the incidence graph of the input instance, which include treewidth  \(\omega \) , clique-width  \(\lambda \) , and diameter  \(\delta \) . The incidence graph is a bipartite graph with one partition of vertices for the children and another for the gifts, where an edge connects a child and a gift if the gift contributes a non-zero happiness value to that child. We show that the problem is W[1]-hard when parameterized by \((\omega ,\lambda ,\delta )\) , even when the incidence graph is planar. We also establish W[1]-hardness under the combined parameters \((\tau ,\lambda ,\delta )\) . These results extend to the Restricted Makespan Minimization Problem, highlighting the inherent difficulty of achieving fair allocations even under strong structural restrictions.