Let \(G\) be a locally compact Abelian group. A discrete subgroup \(D\) is called a uniform lattice if the quotient group \(G/D\) is compact. For such \(D\), let \(D^\perp\) denote the annihilator of \(D\) in the dual group \(\widehat G\) of \(G\) and \(s(D)\) the Haar measure of a fundamental domain for \(D\). Let \((g_i)_{i\in\mathbb{N}}\) be a sequence of functions in \(L^2(G)\) and \((D_i)_{i\in\mathbb{N}}\) a sequence of uniform lattices in \(G\). The authors consider families of functions of the form \[ \Phi^{\{g_i\}}_{\{D_i\}}= \{T_{\lambda_i} g_i: \lambda_i\in D_i,\,i\in\mathbb{N}\}, \] where \(T_xf(y)= f(x^{-1}y)\), \(f\in L^2(G)\), \(x,y\in G\). The main result of the paper states that under a certain local integrability condition, the set \(\Phi^{\{g_i\}}_{\{D_i\}}\) is a Parseval frame for \(L^2(G)\) if and only if for each \(\alpha\in\bigcup^\infty_{i=1} D^\perp_i\) one has \[ \sum{1\over s(D_i)} \overline{\widehat g_i(\omega)}\widehat g_i(\omega\alpha)= \delta_{\alpha, 1} \] for almost all \(\omega\in\widehat G\), where the summation extends over all \(i\in\mathbb{N}\) such that \(\alpha\in D^\perp_i\). The proof essentially consists of an expertly exploitation of the duality theory and the Plancherel formula for the group \(G\). This result allows to treat several classes of reproducing systems on \(L^2(G)\) in a unified manner. The authors demonstrate this in various different situations. They also comment on when the local integrability condition is satisfied.