We consider discrete-event systems consisting of parallel arrays of machines and buffers. The machines are divided into groups in each of which the members have identical structure, i.e. same state set and isomorphic transitions. This feature allows event relabelling of the machines in a given group to a standard prototype machine. In these systems, to avoid the underflow or overflow of the buffers, the controller needs only the information of the total numbers of components at each state and the numbers of workpieces in the buffers. By exploiting the identical structure of each group, we extract such control information from the control functions computed by the state tree structures to generate abstract control functions. Thanks to the symmetry of the system, we show that all controllable events relabelled to the same symbol share an invariant abstract control function, which is independent of the total number of machines, as long as the buffer sizes are fixed. The approach is illustrated by two examples.