We report the first high spatial resolution measurement of magnetic fields surrounding LkH$alpha$ 101, a part of the Auriga-California molecular cloud. The observations were taken with the POL-2 polarimeter on the James Clerk Maxwell Telescope within the framework of the B-fields In Star-forming Region Observations (BISTRO) survey. Observed polarization of thermal dust emission at 850 $mu$m is found to be mostly associated with the red-shifted gas component of the cloud. The magnetic field displays a relatively complex morphology. Two variants of the Davis-Chandrasekhar-Fermi method, unsharp masking and structure function, are used to calculate the strength of magnetic fields in the plane of the sky, yielding a similar result of $B_{rm POS}sim 115$ $mathrm{mu}$G. The mass-to-magnetic-flux ratio in critical value units, $lambdasim0.3$, is the smallest among the values obtained for other regions surveyed by POL-2. This implies that the LkH$alpha$ 101 region is sub-critical and the magnetic field is strong enough to prevent gravitational collapse. The inferred $delta B/B_0sim 0.3$ implies that the large scale component of the magnetic field dominates the turbulent one. The variation of the polarization fraction with total emission intensity can be fitted by a power-law with an index of $alpha=0.82pm0.03$, which lies in the range previously reported for molecular clouds. We find that the polarization fraction decreases rapidly with proximity to the only early B star (LkH$alpha$ 101) in the region. The magnetic field tangling and the joint effect of grain alignment and rotational disruption by radiative torques are potential of explaining such a decreasing trend.
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