Exploiting the sparsity in learning algorithms is a key to achieve excellent performances of adaptive filters. This can be realized by the adaptive proximal forward-backward splitting with carefully chosen parameters. In this paper, we propose an automatic parameter tuning based on a minimization principle of a stochastic approximation of the system-mismatch. The proposed approximation has a Tikhonov-type regularization term, which aims to minimize the disturbance by the update of the adaptive filter and mitigates overfitting to an instantaneous observation. Thanks to these properties, the proposed method realizes adaptive parameter tuning without any user-defined parameters, unlike our previous method that utilizes the user-defined parameter to avoid over-fitting. A numerical example demonstrates the efficacy of the proposed parameter tuning.