Let $(\cx, d, \mu)$ be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of Hytonen. In this paper, we introduce the parametric Marcinkiewicz integral in $(\cx, d, \mu)$. Under the assumption that the parametric Marcinkiewicz integral is bounded on $L^{p_0}(\mu)$ for some $p_0\in(1, \infty)$, then we establish its boundedness from $L^1(\mu)$ to $L^{1,\infty}(\mu)$, from the atomic Hardy space $H^1(\mu)$ to $L^1(\mu)$ and from $L^{\infty}(\mu)$ to $\rblo$ which is a proper subset of ${\rm RBMO}(\mu)$ on $(\cx, d, \mu)$, respectively. As a corollary, we obtain the boundedness of the parametric Marcinkiewicz integral on $L^p(\mu)$ with $p\in{(1,\infty)}$.