The ratio value – known as plasma parameter “beta” (β) - of the interplanetary-plasma pressure (thermal pressure) to the magnetic field’s (magnetic pressure) or of their energy densities, is critical for the space plasmas and has important consequences to their properties. Although in the scientific literature the contribution of the supra-thermal particles to the plasma pressure is generally assumed negligible, we deduced, by analyzing energetic particles and magnetic field measurements recorded by the Ulysses spacecraft, that in a series of events, the energy density contained in the supra-thermal tail of the particle distribution is comparable to or even higher than the energy density of the magnetic field, creating conditions of high-beta plasma. In particular, in this paper we analyze Ulysses measurements of the energy density ratio (parameter βep) of the supra-thermal (20 keV to ~5 MeV) ions’ (by the HI-SCALE instrument) to the magnetic field’s (by the VHM/FGM instrument) in order to find occurrences of high-beta (β ep >1) supra-thermal plasma conditions in the vicinity of interplanetary shock-waves. These high-beta ion events are associated with energetic particle intensity enhancements which have been produced by reacceleration at the shock fronts.