  For the geometry of oriented $(2, 3, 5)$ distributions $(M, {\mathbf D})$,
which correspond to regular, normal parabolic geometries of type
$(\mathrm{G}_2, P)$ for a particular parabolic subgroup $P < \mathrm{G}_2$, we
develop the corresponding tractor calculus and use it to analyze the first BGG
operator $\Theta_0$ associated to the $7$-dimensional irreducible
representation of $\mathrm{G}_2$. We give an explicit formula for the normal
connection on the corresponding tractor bundle and use it to derive explicit
expressions for this operator. We also show that solutions of this operator are
automatically normal, yielding a geometric interpretation of $\ker \Theta_0$:
For any $(M, {\mathbf D})$, this kernel consists precisely of the almost
Einstein scales of the Nurowski conformal structure on $M$ that ${\mathbf D}$
determines.
  We apply our formula for $\Theta_0$ (1) to recover efficiently some known
solutions, (2) to construct a distribution with root type $[3, 1]$ with a
nonzero solution, and (3) to show efficiently that the conformal holonomy of a
particular $(2, 3, 5)$ conformal structure is equal to $\mathrm{G}_2$.
