Using angle resolved photoemission it is shown that the low lying electronic states of the iron pnictide parent compound EuFe$_2$As$_2$ are strongly modified in the magnetically ordered, low temperature, orthorhombic state compared to the tetragonal, paramagnetic case above the spin density wave transition temperature. Back-folded bands, reflected in the orthorhombic/ anti-ferromagnetic Brillouin zone boundary hybridize strongly with the non-folded states, leading to the opening of energy gaps. As a direct consequence, the large Fermi surfaces of the tetragonal phase fragment, the low temperature Fermi surface being comprised of small droplets, built up of electron and hole-like sections. These high resolution ARPES data are therefore in keeping with quantum oscillation and optical data from other undoped pnictide parent compounds. ; Comment: 4 figures, 6 pages