Parametric timed automata (PTA) extend timed automata by allowing parameters in clock constraints. Such a formalism is for instance useful when reasoning about unknown delays in a timed system. Using existing techniques, a user can synthesize the parameter constraints that allow the system to reach a specified goal location, regardless of how much time has passed for the internal clocks. We focus on synthesizing parameters such that not only the goal location is reached, but we also address the following questions: what is the minimal time to reach the goal location? and for which parameter values can we achieve this? We analyse the problem and present an algorithm that solves it. We also discuss and provide solutions for minimizing a specific parameter value to still reach the goal. We empirically study the performance of these algorithms on a benchmark set for PTAs and show that minimal-time reachability synthesis is more efficient to compute than the standard synthesis algorithm for reachability.