  In this article, a logahoric Higgs torsor is defined as a parahoric torsor
with a logarithmic Higgs field. For a connected complex reductive group $G$, we
introduce a notion of stability for logahoric
$\mathcal{G}_{\boldsymbol\theta}$-Higgs torsors on a smooth algebraic curve
$X$, where $\mathcal{G}_{\boldsymbol\theta}$ is a parahoric group scheme on
$X$. In the case when the group $G$ is the general linear group ${\rm GL}_n$,
we show that the stability condition of a parahoric torsor is equivalent to the
stability of a parabolic bundle. A correspondence between semistable logahoric
$\mathcal{G}_{\boldsymbol\theta}$-Higgs torsors and semistable equivariant
logarithmic $G$-Higgs bundles allows us to construct the moduli space
explicitly. This moduli space is shown to be equipped with an algebraic Poisson
structure.
