For a quadrilateral $P$, we consider the centroid $G_0$ of the vertices of $P$, the perimeter centroid $G_1$ of the edges of $P$ and the centroid $G_2$ of the interior of $P$, respectively. We denote by $M$ the intersection point of two diagonals of $P$. If $P$ is a parallelogram, then we have $G_0=G_1=G_2=M$. Conversely, one of $G_0=M$ and $G_2=M$ implies that $P$ is a parallelogram. In this paper, we show that $G_1=M$ is also a characteristic property of parallelograms.