We consider parametric reachability control problems for real-time systems. We model the plant as an extension of parametric timed automata in which the timing constraints on these clocks can make use of parameters. This extension, which we call parametric game automata (PGAs), allows for partitioning the actions in the model between two antagonistic entities: the controller and the environment. The most general problem we study then consists in synthesising both a controller and values for the parameters such that some control location of the automaton is reachable. This problem is undecidable and we therefore provide a subclass of PGA called L/U game automata for which it is decidable. We then consider a backward fixed-point semi-algorithm for solving timed games with reachability objective allowing to compute the most permissive winning strategy. We argue the relevance of this approach and demonstrate its practical usability with a small case-study.