The literature on compressive parameter estimation has been mostly focused on the use of sparsity dictionaries that encode a discretized sampling of the parameter space; these dictionaries, however, suffer from coherence issues that must be controlled for successful estimation. To bypass such issues with discretization, we propose the use of statistical parameter estimation methods within the Approximate Message Passing (AMP) algorithm for signal recovery. Our method leverages the recently proposed use of custom denoisers in place of the usual thresholding steps (which act as denoisers for sparse signals) in AMP. We introduce the design of analog denoisers that are based on statistical parameter estimation algorithms, and we focus on two commonly used examples: frequency estimation and bearing estimation, coupled with the Root MUSIC estimation algorithm. We first analyze the performance of the proposed analog denoiser for signal recovery, and then link the performance in signal estimation to that of parameter estimation. Numerical experiments show significant improvements in estimation performance versus previously proposed approaches for compressive parameter estimation.