Let \(\Omega\) be a smoothly bounded domain in \(\mathbb R^n\), \(p,q>0\), \(pq>1\), \(\alpha:=(p+1)/(pq-1)\), \(\beta:=(q+1)/(pq-1)\). Consider the parabolic system \(u_t=\Delta u+v^p\), \(v_t=\Delta v+u^q\), \(x\in\Omega\), \(t>0\), complemented by the homogeneous Dirichlet boundary conditions and the initial conditions \(u(x,0)=u_0(x)\), \(v(x,0)=v_0(x)\). Assume that \(u_0,v_0\) are nonnegative \(C^1\)-functions vanishing on the boundary of \(\Omega\). Let the corresponding solution \((u,v)\) blow up in finite time \(T\), and let \(u_t,v_t\geq 0\) on \(\Omega\times(0,T)\). The author proves the following assertions. (i) If \(p,q\geq 1\) then there exists a function \(C:\Omega\to\mathbb R\) such that \(u(x,t)\leq C(x)(T-t)^{-\alpha}\) and \(v(x,t)\leq C(x)(T-t)^{-\beta}\) on \(\Omega\times[0,T)\). Moreover, \(C\) does not depend on \(x\) if \(\Omega\) is a ball and \(u_0,v_0\) are radial and radially deacreasing. (ii) If the above upper blow-up rate estimates are true with \(C\) independent of \(x\) then there exists \(c>0\) such that the lower estimates \(\max_\Omega u(x,t)\geq c(T-t)^{-\alpha}\) and \(\max_\Omega v(x,t)\geq c(T-t)^{-\beta}\) are true as well.