Time-varying parameter (TVP) models are widely used in time series analysis for their ability to capture gradual changes in the effect of explanatory variables on an outcome variable of interest. The high degree of flexibility they offer can lead to overfitting when not properly regularized, which in turn results in poor out of sample predictive performance. On the other hand, approaches that are too restrictive risk not letting salient features of the data filter through. In light of these requirements, we propose a novel shrinkage process for sparse state space and TVP models. Building on the work of Cadonna et al. (2020) we leverage the desirable properties of the triple gamma prior and introduce a shrinkage process that aims to combine sufficient regularization with enough flexibility to capture salient features of the data. Links to the work of Kowal et al. (2019) are explored and an efficient MCMC algorithm is discussed.