Objective: We propose a method for the reconstruction of parameter-maps in Quantitative Magnetic Resonance Imaging (QMRI). Methods: Because different quantitative parameter-maps differ from each other in terms of local features, we propose a method where the employed dictionary learning (DL) and sparse coding (SC) algorithms automatically estimate the optimal dictionary-size and sparsity level separately for each parameter-map. We evaluated the method on a $T_1$-mapping QMRI problem in the brain using the BrainWeb data as well as in-vivo brain images acquired on an ultra-high field 7T scanner. We compared it to a model-based acceleration for parameter mapping (MAP) approach, other sparsity-based methods using total variation (TV), Wavelets (Wl) and Shearlets (Sh), and to a method which uses DL and SC to reconstruct qualitative images, followed by a non-linear (DL+Fit). Results: Our algorithm surpasses MAP, TV, Wl and Sh in terms of RMSE and PSNR. It yields better or comparable results to DL+Fit by additionally significantly accelerating the reconstruction by a factor of approximately seven. Conclusion: The proposed method outperforms the reported methods of comparison and yields accurate $T_1$-maps. Although presented for $T_1$-mapping in the brain, our method's structure is general and thus most probably also applicable for the the reconstruction of other quantitative parameters in other organs. Significance: From a clinical perspective, the obtained $T_1$-maps could be utilized to differentiate between healthy subjects and patients with Alzheimer's disease. From a technical perspective, the proposed unsupervised method could be employed to obtain ground-truth data for the development of data-driven methods based on supervised learning.+