  We extend the $(1+1)$-dimensional Dirac-Moshinsky oscillator by changing the
standard derivative by the Dunkl derivative. We demonstrate in a general way
that for the Dirac-Dunkl oscillator be parity invariant, one of the spinor
component must be even, and the other spinor component must be odd, and vice
versa. We decouple the differential equations for each of the spinor component
and introduce an appropriate $su(1,1)$ algebraic realization for the cases when
one of these functions is even and the other function is odd. The
eigenfunctions and the energy spectrum are obtained by using the $su(1,1)$
irreducible representation theory. Finally, by setting the Dunkl parameter to
vanish, we show that our results reduce to those of the standard
Dirac-Moshinsky oscillator.
