It is well known that parallel magnetic field fluctuations are connected with finite plasma beta effects. It can therefore be expected that Reduced Models that simplify or neglect the parallel magnetic field may not capture the salient physics of all types of pressure gradient driven instabilities. This is particularly true in axisymmetric toroidal plasmas in which pressure driven MHD instabilities are always associated with toroidicty and the coupling of the poloidal mode harmonics. This contribution examines the distinct role of parallel magnetic field perturbations on short and long wavelength pressure driven MHD instabilities, and investigates quantitatively the impact of reduced electromagnetic models frequently deployed in fluid and gyrokinetic codes.