  We derive a general expression of the quantum Fisher information for a
Mach-Zehnder interferometer, with the port inputs of an \emph{arbitrary} pure
state and a squeezed thermal state. We find that the standard quantum limit can
be beaten, when even or odd states are applied to the pure-state port. In
particular, when the squeezed thermal state becomes a thermal state, all the
even or odd states have the same quantum Fisher information for given photon
numbers. For a squeezed thermal state, optimal even or odd states are needed to
approach the Heisenberg limit. As examples, we consider several common even or
odd states: Fock states, even or odd coherent states, squeezed vacuum states,
and single-photon-subtracted squeezed vacuum states. We also demonstrate that
super-precision can be realized by implementing the parity measurement for
these states.
