Let \(V\) be an open and bounded subset of \(\mathbb{R}^d\). For each parameter \(t\in \overline{V}\) we consider a conformal iterated function system (IFS) \((f_i(\cdot, t))_{i=1}^k\) in \(\mathbb{R}^d\) depending on the parameter \(t\). By assuming this dependence to be smooth, the author proves: For each \(p\), let \(G_p= \{t\in \overline{V}: \dim_H \Lambda_t\leq p\}\), here \(\Lambda_t\) denotes the limit set of the IFS. If the IFS satisfies the transversality condition and \(p< \min(d,s(u))\), then \[ \limsup_{r\to 0} \dim_p (G_p\cap B_r(u))\leq p, \] where \(s(u)\) denotes the solution of Bowen's equation.