Plug-and-Play priors recently emerged as a powerful technique for solving inverse problems by plugging a denoiser into a classical optimization algorithm. The denoiser accounts for the regularization and therefore implicitly determines the prior knowledge on the data, hence replacing typical handcrafted priors. In this paper, we extend the concept of Plug-and-Play priors to use denoisers that can be parameterized for non-constant noise variance. In that aim, we introduce a preconditioning of the ADMM algorithm, which mathematically justifies the use of such an adjustable denoiser. We additionally propose a procedure for training a convolutional neural network for high quality non-blind image denoising that also allows for pixel-wise control of the noise standard deviation. We show that our pixel-wise adjustable denoiser, along with a suitable preconditioning strategy, can further improve the Plug-and-Play ADMM approach for several applications, including image completion, interpolation, demosaicing and Poisson denoising.