  We consider an extension of the real Lie algebra $\mathfrak{su}(2)$ by
introducing a parity operator $P$ and a parameter $c$. This extended algebra is
isomorphic to the Bannai-Ito algebra with two parameters equal to zero. For
this algebra we classify all unitary finite-dimensional representations and
show their relation with known representations of $\mathfrak{su}(2)$. Moreover,
we present a model for a one-dimensional finite oscillator based on the
odd-dimensional representations of this algebra. For this model, the spectrum
of the position operator is equidistant and coincides with the spectrum of the
known $\mathfrak{su}(2)$ oscillator. In particular the spectrum is independent
of the parameter $c$ while the discrete position wavefunctions, which are given
in terms of certain dual Hahn polynomials, do depend on this parameter.
