Hydrostatic pressure (up to 16 kbar) has been used to investigate the transport properties of GaInAs-AlInAs heterojunctions for the ranges B<or=18 T and 4.2<T<300K. The authors demonstrate that a parallel-conduction process is the cause of the absence of plateaus in the Hall effect at low pressure. The hydrostatic pressure acts on the Si donor level in AlInAs, reducing the free-electron concentration in this layer and the charge transfer to the quantum well. The conductivity of the AlInAs layer becomes negligible at approximately=7 kbar, and this leads to the first observation of quantum Hall plateaus in the system. A simple model based on the triangular-well approximation is used together with low- and high-magnetic-field measurements to extract the concentration of the bulk layers to conduction over the whole of the ranges of pressure and temperature.