Parametric timed automata (PTAs) allow to reason on systems featuring concurrency and timing constraints making use of parameters. Most problems are undecidable for PTAs, including the paramet-ric reachability emptiness problem, i. e., whether at least one parameter valuation allows to reach some discrete state. In this paper, we first exhibit a subclass of PTAs (namely integer-points PTAs) with bounded rational-valued parameters for which the parametric reachability emptiness problem is decidable. Second, we present further results improving the boundary between decidability and undecidability for PTAs and their subclasses.