Summary: The authors obtain the \((L^1,L^{1,\infty})\)-type boundedness for the parametrized Littlewood-Paley operator \(\mu_\lambda^{*,\rho}\) with a kernel satisfying the logarithmic type Lipschitz condition. Moreover, the \(L^p(\mathbb{R}^{n})\)-boundedness of the operator \(\mu_\lambda^{*,\rho}\) can be deduced, where \(1<p<\infty\).