Summary: Consider the wreath product \(G_n=\Gamma^n\rtimes S_n\) of a finite group \(\Gamma \) with the symmetric group \(S_n\). Let \(\Delta_n\) denote the diagonal in \(\Gamma ^n\). Then \(K_n=\Delta_n\times S_n\) forms a subgroup of \(G_n\). In case \(\Gamma \) is abelian, \((G_n,K_n)\) forms a Gelfand pair with relevance to the study of parking functions. For \(\Gamma \) non-abelian it was suggested by K. Aker and M. B. Can [Proc. Am. Math. Soc. 140, No. 4, 1113–1124 (2012; Zbl 1254.20011)] that \((G_n,K_n)\) must fail to be a Gelfand pair for \(n\) sufficiently large. We prove that this is indeed the case: for \(\Gamma \) non-abelian there is some integer \(2 < N(\Gamma)\leq |\Gamma |\) for which \((K_n,G_n)\) is a Gelfand pair for all \(n < N(\Gamma)\) but \((K_n,G_n)\) fails to be a Gelfand pair for all \(n\geq N(\Gamma)\). For dihedral groups \(\Gamma =D_p\) with \(p\) an odd prime we prove that \(N(\Gamma)=6\).