When modeling systems and designing digital twins, we often rely on indirect observations simulated as the solution of partial differential equations whose parameters depend non-linearly on the physical properties of the system. Whereas solving these equations is doable using finite element approaches, probabilistic inversion, where the physical properties of the elements of the discretization are described by probability distributions, is needed instead of deterministic methods to quantify the uncertainty. However, the high dimensionality of the parameter space to be inferred results in unaffordable computational times. Adaptive parametrization, using structures of lower dimensionality than the finite element mesh, presents a more efficient solution. This alternative has been used in seismic tomography to infer both the number and the physical properties of geological cells \cite{Paper}, but it has not been widely implemented simultaneously with finite element meshes. In this work, we test and discuss how adaptive parametrization strategies in Bayesian inversions work compared to regular parametrization methods in a Poisson problem. We assign a specific structure to its domain and perform Markov chain Monte Carlo (MCMC) inversions to recover the thermal conductivity as a probability distribution based on the likelihood of the temperature measurements. As a starting point, we parametrize the physical properties of the subsurface domain equal to the high-dimensional finite element grid. We use adaptive MCMC techniques to accelerate convergence, reduce the risk of getting trapped in local minima, and determine the optimal meta parameters on the run. Then, we use a new parametrization based on the assigned structure and take the number of parameters as an unknown itself. Our results show that we recover the parameters structure in all cases but with improved performance when using transdimensionality, addressing the lack of information on subsurface heterogeneity by modifying the number of parameters locally.