Before revision:  We provide a novel characterization: the parallel trends assumption holds under all strictly monotonic transformations of the outcome if and only if a stronger "parallel trends"-type condition holds on the entire distribution of untreated potential outcomes.
After revision:  We provide a novel characterization: the parallel trends assumption holds under all strictly monotonic transformations of the outcome if and only if a stronger "parallel trends"-type condition holds for the cumulative distribution function of untreated potential outcomes.