An ionization front (IF) surrounding an H II region is a sharp interface where a cold neutral gas makes transition to a warm ionized phase by absorbing UV photons from central stars. We investigate the instability of a plane-parallel D-type IF threaded by parallel magnetic fields, by neglecting the effects of recombination within the ionized gas. We find that weak D-type IFs always have the post-IF magnetosonic Mach number $mathcal{M}_{rm M2} leq 1$. For such fronts, magnetic fields increase the maximum propagation speed of the IFs, while reducing the expansion factor $alpha$ by a factor of $1+1/(2beta_1)$ compared to the unmagnetized case, with $beta_1$ denoting the plasma beta in the pre-IF region. IFs become unstable to distortional perturbations due to gas expansion across the fronts, exactly analogous to the Darrieus-Landau instability of ablation fronts in terrestrial flames. The growth rate of the IF instability is proportional linearly to the perturbation wavenumber as well as the upstream flow speed, and approximately to $alpha^{1/2}$. The IF instability is stabilized by gas compressibility and becomes completely quenched when the front is D-critical. The instability is also stabilized by magnetic pressure when the perturbations propagate in the direction perpendicular to the fields. When the perturbations propagate in the direction parallel to the fields, on the other hand, it is magnetic tension that reduces the growth rate, completely suppressing the instability when $mathcal{M}_{rm M2}^2 < 2/(beta_1 - 1)$. When the front experiences an acceleration, the IF instability cooperates with the Rayleigh-Taylor instability to make the front more unstable.
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