A significant fraction of OB-type, main-sequence massive stars are classified as runaway and move supersonically through the interstellar medium (ISM). Their strong stellar winds interact with their surroundings where the typical strength of the local ISM magnetic field is about 3.5-7 µG, which can result in the formation of bow shock nebulae. We investigate the effects of such magnetic fields, aligned with the motion of the flow, on the formation and emission properties of these circumstellar structures. Our axisymmetric, magneto-hydrodynamical simulations with optically-thin radiative cooling, heating and anisotropic thermal conduction show that the presence of the background ISM magnetic field affects the projected optical emission our bow shocks at Hα and [OIII] λ 5007 which become fainter by about 1-2 orders of magnitude, respectively. Radiative transfer calculations against dust opacity indicate that the magnetic field slightly diminishes their projected infrared emission and that our bow shocks emit brightly at 60 µm. This may explain why the bow shocks generated by ionizing runaway massive stars are often difficult to identify. Finally, we discuss our results in the context of the bow shock of ζ Ophiuchi and we support the interpretation of its imperfect morphology as an evidence of the presence of an ISM magnetic field not aligned with the motion of its driving star.