  This paper studies the minimum weight set cover (MinWSC) problem with a {\em
small neighborhood cover} (SNC) property proposed by Agarwal {\it et al.} in
\cite{Agarwal.}. A parallel algorithm for MinWSC with $\tau$-SNC property is
presented, obtaining approximation ratio $\tau(1+3\varepsilon)$ in
$O(L\log_{1+\varepsilon}\frac{n^3}{\varepsilon^2}+ 4\tau^{3}2^\tau L^2\log n)$
rounds, where $0< \varepsilon <\frac{1}{2}$ is a constant, $n$ is the number of
elements, and $L$ is a parameter related to SNC property. Our results not only
improve the approximation ratio obtained in \cite{Agarwal.}, but also answer
two questions proposed in \cite{Agarwal.}.
