We consider probabilistic timed automata (PTA) in which probabilities can be parameters, i.e. symbolic constants. They are useful to model randomised real-time systems where exact probabilities are unknown, or where the probability values should be optimised. We prove that existing techniques to transform probabilistic timed automata into equivalent finite-state Markov decision processes (MDPs) remain correct in the parametric setting, using a systematic proof pattern. We implemented two of these parameter-preserving transformations-using digital clocks and backwards reachability-in the Modest Toolset. Using Storm's parameter space partitioning approach, parameter values can be efficiently synthesized in the resulting parametric MDPs. We use several case studies from the literature of varying state and parameter space sizes to experimentally evaluate the performance and scalability of this novel analysis trajectory for parametric PTA.