In the present article, we introduce a new class of operators which will be called the class of $k$-quasi $*$-paranormal operators that includes $*$-paranormal operators. A part from other results, we show that following results hold for a $k$-quasi $*$-paranormal operator $T$: (i) $T$ has the SVEP. (ii) Every non-zero isolated point in the spectrum of $T$ is a simple pole of the resolvent of $T$. (iii) All Weyl type theorems hold for $T$. (iv) Comments and some open problems are also presented.