  The polarized Galactic thermal dust emission constitutes a major probe for
the study and the characterization of the Galactic Magnetic Field (GMF). In
this paper, we apply the maximum-likelihood analysis that we established in our
companion paper (Pelgrims, Mac\'ias-P\'erez & Ruppin) to model the large-scale
regular-component of the GMF from the polarized diffuse emission from Galactic
thermal dust as measured by $Planck$ at 353 GHz. As a first attempt, we
consider three models to describe the dust density distribution across the
whole Galaxy and four models for the GMF. All models are parametric and
heuristics and leave us with twelve reconstructions of the GMF geometrical
structure. These reconstructions are obtained at the $N_{\rm{side}} = 64$ and
provide the first constraints on the GMF obtained through Markov Chain Monte
Carlo analysis from thermal dust polarization data. This work demonstrates that
competitive constraints on the GMF can be obtained from the polarized thermal
dust sky as compared to other observational probes.
