  With the rapid increase of fast radio burst (FRB) detections within the past
few years, there is now a catalogue being developed for all-sky extragalactic
dispersion measure (DM) observations in addition to the existing collection of
all-sky extragalactic Faraday rotation measurements (RMs) of radio galaxies. We
present a method of reconstructing all-sky information of the Galactic magnetic
field component parallel to the line of sight, $B_{\parallel}$, using simulated
observations of the RM and DM along lines of sight to radio galaxies and FRB
populations, respectively. This technique is capable of distinguishing between
different input Galactic magnetic field and thermal electron density models.
Significant extragalactic contributions to the DM are the predominant
impediment in accurately reconstructing the Galactic DM and
$\left<B_{\parallel}\right>$ skies. We look at ways to improve the
reconstruction by applying a filtering algorithm on the simulated DM lines of
sight and we derive generalized corrections for DM observations at $|b|$ > 10
deg that help to disentangle Galactic and extragalactic DM contributions.
Overall, we are able to reconstruct both large-scale Galactic structure and
local features in the Milky Way's magnetic field from the assumed models. We
discuss the application of this technique to future FRB observations and
address possible differences between our simulated model and observed data,
namely: adjusting the priors of the inference model, an unevenly distributed
population of FRBs on the sky, and localized extragalactic DM structures.
