We investigate both analytically and numerically, how the magnetic counter-streaming instability in unmagnetized pair (electron-positron) plasmas is influenced by the external magnetic field parallel to the streaming direction. It is shown from the linear theory of relativistic cold four fluids model that the generation of quasi-static magnetic field whose wave vector is perpendicular to the streaming direction is restrained by the increasing external magnetic field. The linear theory is also confirmed by the particle-in-cell (PIC)simulation. We show that the counter-streaming instability changes its character from magnetic to electrostatic nature when the external magnetic field increases as 1.5 ω_pe < ω_ce. The electrostatic waves growing due to the electrostatic counter-streaming instability play an important role for producting fast electrons and positrons with energy of MeV. The process of high-energy particle production in relativistic shocks in magnetized pair plasmas may be applied to gamma-ray burst events.