  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 Poeschl--Teller potentials of
this kind.
