For lack of a better term and for this purpose only, let us call a refinement $\mathcal{V}$ of a cover $\mathcal{U}$ faithful if $\mathcal{V}$ can be written as $\mathcal{V}=\{V_U:U\in\mathcal{U}\}$ where $V_U\subseteq U$ for each $U\in\mathcal{U}$.
Does every open cover of a paracompact space have a faithful open refinement that is locally finite?
In other words (and maybe clearer): if, for every open cover $\mathcal{U}$ of a topological space, we demand the existence of such an open and locally finite cover $\{V_U:U\in\mathcal{U}\}$ that $V_U\subseteq U$ for each $U\in\mathcal{U}$, do we get an equivalent or strictly stronger notion than paracompactness?