We have obtained sensitive dust continuum polarization observations at 850 μm in the B213 region of Taurus using POL-2 on SCUBA-2 at the James Clerk Maxwell Telescope as part of the B-fields in STar-forming Region Observations (BISTRO) survey. These observations allow us to probe magnetic field (B-field) at high spatial resolution (∼2000 au or ∼0.01 pc at 140 pc) in two protostellar cores (K04166 and K04169) and one prestellar core (Miz-8b) that lie within the B213 filament. Using the Davis–Chandrasekhar–Fermi method, we estimate the B-field strengths in K04166, K04169, and Miz-8b to be 38 ± 14, 44 ± 16, and 12 ± 5 μG, respectively. These cores show distinct mean B-field orientations. The B-field in K04166 is well ordered and aligned parallel to the orientations of the core minor axis, outflows, core rotation axis, and large-scale uniform B-field, in accordance with magnetically regulated star formation via ambipolar diffusion taking place in K04166. The B-field in K04169 is found to be ordered but oriented nearly perpendicular to the core minor axis and large-scale B-field and not well correlated with other axes. In contrast, Miz-8b exhibits a disordered B-field that shows no preferred alignment with the core minor axis or large-scale field. We found that only one core, K04166, retains a memory of the large-scale uniform B-field. The other two cores, K04169 and Miz-8b, are decoupled from the large-scale field. Such a complex B-field configuration could be caused by gas inflow onto the filament, even in the presence of a substantial magnetic flux.