  Recently, it was observed that self-interacting scalar quantum field theories
having a non-Hermitian interaction term of the form $g(i\phi)^{2+\delta}$,
where $\delta$ is a real positive parameter, are physically acceptable in the
sense that the energy spectrum is real and bounded below. Such theories possess
PT invariance, but they are not symmetric under parity reflection or time
reversal separately. This broken parity symmetry is manifested in a nonzero
value for $<\phi>$, even if $\delta$ is an even integer. This paper extends
this idea to a two-dimensional supersymmetric quantum field theory whose
superpotential is ${\cal S}(\phi)=-ig(i\phi)^{1+\delta}$. The resulting quantum
field theory exhibits a broken parity symmetry for all $\delta>0$. However,
supersymmetry remains unbroken, which is verified by showing that the
ground-state energy density vanishes and that the fermion-boson mass ratio is
unity.
