a b s t r a c tType A N -fold supercharge admits a one-parameter family of factorizations into product of N first-order linear differential operators due to an underlying GLð2; CÞ 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 ¼ 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.