We present a numerical scheme for the solution of optimal control problems associated with production-destruction systems (PDS). We start by introducing these differential systems and their properties, such as positivity and conservativity. Then, we define a class of controlled PDS in a way that preserves the properties mentioned earlier and we formulate the related general finite horizon optimal control problem. Following the Dynamic Programming approach, the problem is addressed through the solution of an evolutive partial differential equation in the state-space of the PDS, called the Hamilton--Jacobi--Bellman (HJB) equation. We devise a parallel-in-space semi-Lagrangian (SL) scheme for the solution of the HJB equation based on modified-Patankar methods, which preserve positivity and conservativity also at the numerical level, thus obtaining the optimal feedback controls and trajectories. Finally, we show with two examples, enzyme-catalyzed biochemical reactions and infectious diseases, that the proposed scheme provides better solutions compared with a classical SL scheme.