  We report infrared studies of AFe$_{2}$As$_{2}$ (A = Ba, Sr), two
representative parent compounds of iron-arsenide superconductors, at magnetic
fields (B) up to 17.5 T. Optical transitions between Landau levels (LLs) were
observed in the antiferromagnetic states of these two parent compounds. Our
observation of a $\sqrt{B}$ dependence of the LL transition energies, the
zero-energy intercepts at B = 0 T under the linear extrapolations of the
transition energies and the energy ratio ($\sim$ 2.4) between the observed LL
transitions, combined with the linear band dispersions in two-dimensional (2D)
momentum space obtained by theoretical calculations, demonstrates the existence
of massless Dirac fermions in antiferromagnetic BaFe$_{2}$As$_{2}$. More
importantly, the observed dominance of the zeroth-LL-related absorption
features and the calculated bands with extremely weak dispersions along the
momentum direction $k_{z}$ indicate that massless Dirac fermions in
BaFe$_{2}$As$_{2}$ are 2D. Furthermore, we find that the total substitution of
the barium atoms in BaFe$_{2}$As$_{2}$ by strontium atoms not only maintains 2D
massless Dirac fermions in this system, but also enhances their Fermi velocity,
which supports that the Dirac points in iron-arsenide parent compounds are
topologically protected.
