The influence of magnetic fields (B-fields) on the formation and evolution of bipolar bubbles, due to the expanding ionization fronts (I-fronts) driven by the HII regions that are formed and embedded in filamentary molecular clouds, has not been well-studied yet. In addition to the anisotropic expansion of I-fronts into a filament, B-fields are expected to introduce an additional anisotropic pressure, which might favor the expansion and propagation of I-fronts forming a bipolar bubble. We present results based on near-infrared polarimetric observations toward the central ~8'x8' area of the star-forming region RCW 57A, which hosts an HII region, a filament, and a bipolar bubble. Polarization measurements of 178 reddened background stars, out of the 919 detected sources in the JHKs bands, reveal B-fields that thread perpendicularly to the filament long axis. The B-fields exhibit an hourglass morphology that closely follows the structure of the bipolar bubble. The mean B-field strength, estimated using the Chandrasekhar-Fermi method (CF method), is 91+/-8{mu}G. B-field pressure dominates over turbulent and thermal pressures. Thermal pressure might act in the same orientation as the B-fields to accelerate the expansion of those I-fronts. The observed morphological correspondence among the B-fields, filament, and bipolar bubble demonstrate that the B-fields are important to the cloud contraction that formed the filament, to the gravitational collapse and star formation in it, and in feedback processes. The last one includes the formation and evolution of mid- infrared bubbles by means of B-field supported propagation and expansion of I-fronts. These may shed light on preexisting conditions favoring the formation of the massive stellar cluster in RCW 57A.