In this paper, we shall introduce a new class absolute-$*$-$k$-paranormal operators given by a norm inequality and $*$-$A(k)$ operator by operator inequality, we will discuss the inclusion relation of them. And we study spectral properties of class absolute-$*$-$k$-paranormal operators. We show that if $T$ belongs to class absolute -$*$-$k$-paranormal operators, then its point spectrum and joint point spectrum are identical, its approximate point spectrum and joint approximate point spectrum are identical. Next as an application of them, for Weyl spectrum $w(\cdot)$ and essential approximate point spectrum $\sigma_{ea}(\cdot)$, we will show that if $T$ or $T^*$ absolute -$*$-$k$-paranormal for $0\leq k\leq 1$, then $w(f(T))=f(w(T))$, $\sigma_{ea}(f(T))=f(\sigma_{ea}(T))$ for every $f\in H(\sigma (T))$ where $H(\sigma (T))$ denotes the set of all analytic functions on an open neighborhood of $\sigma (T)$.