This work proposes a structured prior integrated within the Bayesian framework for variable flip angle \(T_1\) mapping. The proposed structured prior combines total variation (TV) and \(\ell _1\) norm functions, and is proven to be a proper prior. The TV– \(\ell _1\) prior promotes sparsity in the spatial gradients of the parametric maps, resulting in smooth and coherent image reconstructions. Embedding the prior within the Bayesian framework enables uncertainty quantification for both \(T_1\) and \(M_0\) estimates. Posterior inference was performed using the No-U-Turn Sampler (NUTS). The proposed method is compared to maximum likelihood estimation and to alternative Bayesian models that employ uniform, Laplace, and bounded TV priors. The results show that the proposed method yields narrower probability density functions, indicating reduced uncertainty. The proposed method also achieves lower variance and exhibits a smaller negative bias, reflecting more stable estimates. Overall, the integration of TV and \(\ell _1\) functions in a prior within the Bayesian framework enhances spatial coherence in \(T_1\) mapping and delivers improved uncertainty quantification, making it a promising tool for robust quantitative MRI parameter estimation.