Type A N-fold supercharge admits a one-parameter family of factorizations into product of N first-order linear differential operators due to an underlying GL (2, C) symmetry. As a consequence, a type A N-fold supersymmetric system can have different intermediate Hamiltonians corresponding to different factorizations. We derive the necessary and sufficient conditions for the latter system to possess intermediate Hamiltonians for the N = 2 case. We then show that whenever it has (at least) one intermediate Hamiltonian, it can admit second-order parasupersymmetry and a generalized 2-fold superalgebra. As an illustration, we construct a set of generalized Pöschl-Teller potentials of this kind. © 2009 Elsevier Inc. All rights reserved.