Decidability results for hybrid automata often exploit subtle properties about dimensionality (number of continuous variables), and interaction between discrete transitions and continuous trajectories of variables. Thus, the decidability results often do not carry over to parallel compositions of hybrid automata, even when there is no communication other than the implicit synchronization of time. In this paper, we show that the reachability problem for concurrently running planar, monotonic hybrid automata is decidable. Planar, monotonic hybrid automata are a special class of linear hybrid automata with only two variables, whose flows in all states are simultaneously monotonic along some direction in the plane. The reachability problem is known to be decidable for this class. Our concurrently running automata synchronize with respect to a global clock, and through labeled discrete transitions. The proof of decidability hinges on a new observation that the timed trace language of a planar monotonic automaton can be recognized by a timed automaton, and the decidability result follows from the decidability of composition of timed automata. Our results identify a new decidable subclass of multi-rate hybrid automata.