An ionization front (IF) surrounding an H II region is a sharp interface through which a cold neutral gas makes transition to a warm ionized phase by absorbing UV photons from central massive stars. We investigate the structure and instability of a plane-parallel D-type IF threaded by magnetic fields parallel to the front. We find that magnetic fields increase the maximum propagation speed of the IFs, while reducing the expansion factor, defined as the density ratio of neutral to ionized phases. 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. 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 ${eta}$ < 1.5, with ${eta}$ denoting the square of the ratio of the sound speed to the Alfven speed in the pre-IF region. When the front experiences an acceleration, the IF instability cooperates with the Rayleigh-Taylor instability to make the front more unstable. We discuss potential effects of IF instability on the evolution and dynamics of IFs in the interstellar medium.