When modeling systems and designing digital twins, we often rely on indirect and noise observations. These quantities can be simulated as the solution of partial differential equations whose parameters depend non-linearly on the physical properties of the system under study. Whereas solving these equations is doable using finite element approaches, deterministic methods to perform the inversion do not account for the uncertainty quantification and may lead to unrepresentative solutions, which is why it is preferred to use probabilistic inversion approaches where the physical properties of the elements of the discretization are described by probability distributions. However, the high dimensionality of the parameter space to be inferred results in unaffordable computational times. Moreover, most physical systems present structures of lower dimensionality than the mesh used to solve the forward problem, which hints towards more fit and efficient parametrizations. Adaptive parametrization has been used in seismic tomography to infer both the number and the physical properties of geological cells, 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, solving for temperature and with thermal conductivity as a parameter. We assign to its domain a specific structure that mimics the one of Earth's upper mantle 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 to reduce the dimensionality of the problem 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. Moreover, this latter approach addresses the lack of information on subsurface heterogeneity by modifying the number of parameters locally.