Variational methods are an effective model-based approach for solving data reconstruction inverse problems in many applicative fields such as, e.g., astronomical, seismic, and medical imaging. However, their performance strongly depends on a suitable selection of the values of the free parameters entering the variational model. This selection problem is thus crucial and has been addressed widely for several decades, starting from the simplest case of single-parameter models. One of the approaches still most used today in this case is the discrepancy principle (DP), which relies on imposing a value of the data term approximately equal to its expected value. However, the DP has two severe limitations, namely (1) it requires having a quite precise knowledge of the variance of corrupting noise and (2) it can not be applied successfully to the selection of more than one parameter. Quite recently, it has been proposed another selection criterion called the residual whiteness principle (RWP), which not only outperforms the DP but also does not require knowledge about the noise variance. The idea is to select the parameter value yielding reconstructions with associated residual data which most resemble the realization of a white noise random process. This effective criterion, however, has so far been applied to the selection of only one parameter and only in the case of white measurement noise. In this talk, we extend the RWP by addressing the problem to estimate multiple parameters in variational models for inverse problems subject to both white and non-white but whitenable noise corruptions, thus covering most of the application cases. The presented parameter selection criterion, referred to as the Generalized Whiteness Principle (GWP), is formulated as a bilevel optimization problem. To circumvent the non-smoothness of the variational models typically employed in inverse problems - the non-smoothness representing a bottleneck in the bilevel set-up — we propose to adopt a derivative-free optimization algorithm for the solution of the designed bilevel problem. We refer to this novel numerical solution paradigm as bilevel derivative-free approach. Numerical tests highlight both the ability of the proposed GWP to effectively select multiple parameters and the significant advantages, in terms of computational cost, of the bilevel derivative-free numerical solution framework.