Suppose φ and χ are fields such that, if x and y are spacelike-separated points and i and j represent the spinor/tensor indices, Also suppose χ is invariant under the Z2 parity (nothing to do with spatial reflections!)
Free field theories always satisfy this property.
Then, the Z2 parity of the number of χ particles is well defined and is conserved in time.
The fields φ and χ above don't have the proper statistics relations for either a boson or a fermion.
Assume you have a Z2 conserved parity operator Kχ acting upon χ alone.