Given k terminal pairs and two tuples of (internally) vertex-disjoint shortest paths in an unweighted graph, we study the problem of determining whether one tuple can be transformed into the other by repeatedly replacing a single vertex in one shortest path, while maintaining a tuple of vertex-disjoint shortest paths at every step. In this paper, we analyze the problem from the viewpoint of parameterized complexity. We first provide a tight complexity classification based on the maximum terminal distance D, defined as the largest distance among the terminal pairs: the problem is (para-)PSPACE-complete for any fixed \(D \ge 3\) , while it is solvable in polynomial time when \(D \le 2\) . We then show that the problem is fixed-parameter tractable when parameterized by \(\ell + \varDelta \) , where \(\ell \) is the number of transformation steps and \(\varDelta \) is the maximum degree of an input graph. Finally, we examine the effect of structural graph parameters, by showing that the problem is fixed-parameter tractable when parameterized by k plus one of the modular-width, cluster deletion number, and tree-depth of an input graph.