This paper considers a parametric approach to infer sparse networks described by nonlinear ARX models, with linear ARX treated as a special case. The proposed method infers both the Boolean structure and the internal dynamics of the network. It considers classes of nonlinear systems that can be written as weighted (unknown) sums of nonlinear functions chosen from a fixed basis dictionary. Due to the sparse topology, coefficients of most groups are zero. Besides, only a few nonlinear terms in nonzero groups contribute to the internal dynamics. Therefore, the identification problem should estimate both group- and element-sparse parameter vectors. The proposed method combines Sparse Bayesian Learning (SBL) and Group Sparse Bayesian Learning (GSBL) to impose both kinds of sparsity. Simulations indicate that our method outperforms SBL and GSBL when these are applied alone. A linear ring structure network also illustrates that the proposed method has improved performance to the kernel approach.