  We prove a general formula that relates the parity of the Langlands parameter
of a conjugate self-dual discrete series representation of $\mathrm{GL}_n$ to
the parity of its Jacquet-Langlands image. It gives a generalization of a
partial result by Mieda concerning the case of invariant $1/n$ and
supercuspidal representations. It also gives a variation of the result on the
self-dual case by Prasad and Ramakrishnan.
