Let $(u, v)$ be a nonnegative solution to the semilinear parabolic system $$ \mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_{t} u = D_{1} \Delta u + v^{p}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ \partial_{t} v = D_{2} \Delta v + u^{q}, \quad x \in \mathbf{R}^{N}, \ t > 0,\\ (u(\cdot,0), v(\cdot,0)) = (\mu, \nu), \quad x \in \mathbf{R}^{N}, \end{array} \right. $$ where $D_{1}$, $D_{2} > 0$, $0 < p \leq q$ with $pq > 1$ and $(\mu, \nu)$ is a pair of nonnegative Radon measures or nonnegative measurable functions in $\mathbf{R}^{N}$. In this paper we study sufficient conditions on the initial data for the solvability of problem (P) and clarify optimal singularities of the initial functions for the solvability.