In this paper, we investigate the in-plane stability and post-buckling response of deep parabolic arches with high slenderness ratios subjected to a concentrated load on the apex, using the finite element implementation of a geometrically exact rod model and the cylindrical version of the arc-length continuation method enabled with pivot-monitored branch-switching. The rod model used here includes geometrically exact kinematics of the cross-section, exact kinetics, and a linear elastic constitutive law; and advantageously employs quaternion parameters to treat the cross-sectional rotations and to compute the exponential map in the configurational update of rotations. The evolution of the Frenet frame along the centroidal curve is used to determine the initial curvature of the rod. Three sets of boundary conditions, i.e. fixed-fixed (FF), fixed-pinned (FP) and pinned-pinned (PP), are considered, and arches with a wide range of rise-to-span ratios are analyzed for each set. Complete post-buckling response has been derived for all cases. The results reveal that although all the PP arches and all the FF arches (with the exception of FF arches with rise-to-span ratio less than 0.3) considered in this study buckle via bifurcation, the nature of stability of the different solution branches in the post-buckling regime differs from case to case. All FP slender parabolic arches exhibit limit-point buckling, again with several markedly different post-buckling behaviors. Also, some arches in the FF and PP case are shown to exhibit a clear bistable behavior in the post-buckled state. © 2018 World Scientific Publishing Company.