  We consider the analytic construction of three-point functions of conserved
higher-spin supercurrents in three-dimensional $\mathcal{N}=1$ superconformal
field theory which are Grassmann-odd in superspace. In particular, these
include the three-point functions of the supercurrent and flavour currents,
which contain the three-point functions of the energy-momentum tensor and
conserved vector currents at the component level. We present an analytic proof
for arbitrary superspins that these correlators do not possess a
parity-violating contribution. We also prove that the parity-even contribution
is unique, and exists (under an assumption that is well supported by the
computational approach of arXiv:2302.00593) for arbitrary superspins. The
construction of the parity-even sector is shown to reduce to solving a system
of linear homogeneous equations with a tri-diagonal matrix of co-rank one,
which we solve explicitly for arbitrary superspins.
