As Ben wrote, the question appears to conflate two different parabolic geometries of type $\newcommand{bfD}{{\bf D}}\newcommand{bfE}{{\bf E}}\newcommand{bfH}{{\bf H}}G_2$:

Let $\Bbb V$ be the standard (i.e., $7$-dimensional irreducible) representation of $\mathfrak{g}_2$ (either the split real or the complex form); recall that $G_2$ is the stabilizer of cross product map $\times : \Bbb V \times \Bbb V \to \Bbb V$. The inclusion $G_2 \hookrightarrow SO(3, 4)$ ($G_2 \hookrightarrow SO(7, \Bbb C)$) determines an indefinite, nondegenerate, symmetric bilinear form $H$ on $\Bbb V$.

The "first" parabolic subgroup $P_1$ (corresponding to a cross on the first node of the Dynkin diagram of $G_2$ in the usual Bourbaki ordering) is the stabilizer of an isotropic $1$-dimensional subspace of $\Bbb V$. The cone $\mathcal C$ of nonzero isotropic vectors inherits an invariant filtration of tangent distributions, whose fibers at $Y \in \mathcal C$ are $$\ker (Z \mapsto Y \times Z) \subset \operatorname{im} (Z \mapsto Y \times Z) \subset T_Y \mathcal C$$ of dimensions $3, 4, 6$ (to apply the cross product, we implicitly use here the identifications determined by the canonical isomorphism $T_Y \Bbb V \leftrightarrow \Bbb V$). By linearity, this descends to a filtration $$\bfD \subset \bfD' \subset T\Bbb Q_5,$$ of dimensions $2, 3, 5$, on the null quadric $\Bbb{Q}_5 := \Bbb P(\mathcal C) \subset \Bbb P(\Bbb V)$, which is diffeomorphic to $(\Bbb S^2 \times \Bbb S^3) / \Bbb Z_2$, where $\Bbb Z_2$ acts by the antipodal map on both factors. Since the ingredients are $G_2$-invariant, so is $\bfD$ under the induced action on $T\Bbb Q_5$, and as one expects, it turns out that $[\bfD, \bfD] = \bfD'$ and $[\bfD', \bfD] = T\Bbb Q_5$. (NB there are no $G_2$-invariant linear or hyperplane distributions on $T\Bbb Q_5$.)

On the other hand, consider differential equations of the form $z' = F(x, y, y', y'', z)$. Any function $F(x, y, p, q, z)$ determines a total derivative $D_x := \partial_x + p \partial_y + q \partial_p + F \partial_z$ on the corresponding partial jet space $J^{2, 0}(\Bbb R, \Bbb R) \cong \Bbb R^5_{xypqz}$. Suitably regarded, the vertical fibers of the jet truncation map $J^{2, 0}(\Bbb R, \Bbb R) \to J^{1, 0}(\Bbb R, \Bbb R)$ are spanned by $\partial_q$. Computing directly shows that the distribution $$\bfD_F := \langle D_x, \partial_q \rangle$$ is generic iff $F_{qq}$ vanishes nowhere; conversely, a theorem of (I believe) Monge states than any generic $2$-plane distribution on a $5$-manifold is locally equivalent to $\bfD_F$ for some function $F$. If $F(x, y, p, q, z) := q^2$, then the resulting distribution has infinitesimal symmetry algebra isomorphic to $\mathfrak{g}_2$, so by a general fact about parabolic geometries the distribution $(J^{2, 0}(\Bbb R, \Bbb R), \bfD_F)$ corresponding to the differential equation $z' = (y'')^2$ is everywhere locally diffeomorphic to the homogeneous model distribution $(\Bbb Q_5, \bfD)$ above. In particular, it follows from this that *neither of the distributions $V$ and $\widetilde{C}$ are invariant under the action of the infinitesimal symmetry algebra $\mathfrak{g}_2$ of $(J^{2, 0}(\Bbb R, \Bbb R), \bfD_F)$*.

The correct statement of the theorem you mention is that there is an equivalence of categories between generic $2$-plane distributions on $5$-manifolds and *normal, regular* parabolic geometries of type $(G_2, P_1)$. (See the end of Subsubsection 4.3.2 in Cap & Slovak's book, *Parabolic Geometries*.)

On the other hand, we can consider the action of $G_2$ on the space of isotropic $2$-planes in $\Bbb V$. This action has two orbits, according to whether the cross product $\times$ restricts to the zero map on each $2$-plane. (Bryant calls the $2$-planes on which the restriction is zero *special* in his highly enjoyable lecture notes *Elie Cartan and Geometric Duality* [pdf], which treats the correspondence space construction for $G_2 / P_1 \leftarrow G_2 / (P_1 \cap P_2) \to G_2 / P_2$, as well as the analogous construction for $A_2$ and $B_2 \cong C_2$. NB that this article seems contains a few typos, replacing $\Bbb N_5$, introduced in a moment, with $\Bbb Q_5$, which is the essential apparent confusion in the question here.) The isotropy subgroup of a point in the $5$-dimensional space $\Bbb N_5$ of special $2$-planes is the "second" parabolic $P_2 \subset G_2$. Analogously to the situation for the first parabolic, we can view $\Bbb N_5$ as a subset of $\Bbb P^{13} = \Bbb P(\mathfrak{g}_2)$, but its geometry is apparently much more complicated than that of $\Bbb Q_5$: In the complex case, $\Bbb N_5$ is a variety of degree $18$, and its complete intersection with three hyperplanes in a general configuration is a K3 surface of genus $10$, but NB other geometric descriptions of this space (which look less daunting to non-algebraic geometers like myself) are available, too. Apparently this is worked out in the paper of Borcea cited below, but I can't find an ungated copy. See also the accessible historical survey paper of Agricola, also cited below.

Now, $\Bbb N_5$ inherits an invariant contact distribution $\bfH$. (Surely this can be written down with some much effort in terms of the cross product on $\Bbb V$, but to my knowledge this hasn't been done anywhere.) Moreover, the representation $P_2$ induces on each fiber of $\bfH$ turns out to be a trivial extension of a representation of $GL(2, \Bbb F) \subset P_2$, and this representation is isomorphic to $S^3 \Bbb F^2$ (this representation is conformally symplectic and so determines equivalently a nondegenerate cone in each fiber of $\bfH$). All of the $G_2$-invariant structure on $\Bbb N_5$ can be recovered from these objects, corresponding to the fact that a (normal, regular) parabolic geometry of type $(G_2, P_2)$ is a $5$-manifold $M$ equipped with a "$G_2$ contact structure", which is a contact structure $\bfH \subset TM$ together with an auxiliary rank-$2$ vector bundle $\bfE \to M$ and a vector bundle isomorphism $S^3 \bfE \stackrel{\cong}{\to} \bfH$ such that the Levi bracket $\bfH \times \bfH \to TM / \bfH$ (the tensorial map induced by the Lie bracket) is invariant under the induced action of $\mathfrak{sl}(\bfE)$. See Subsubsection 4.2.8 of Cap & Slovak's book. I don't know of any sensible analog of the Monge (quasi-)normal form $z' = F(\cdots)$ for $G_2$ contact structures, but I would be pleased to hear about one.

There are connections between these two types of parabolic geometry beyond the mentioned correspondence space construction. See this preprint of Leistner, Nurowski, & Sagerschnig.

I. Agricola, Old and New on the Exceptional Group $G_2$ [pdf], *Notices Amer. Math. Soc.* **55**(8) (2008), 922-929.

C. Borcea, Smooth global complete intersections in certain compact homogeneous complex manifolds, *J. Reine Angew. Math.* **344** (1983), 65–70.

A. Cap, J. Slovak, *Parabolic geometries I: Background and general theory*. Math. Surveys Monogr. **154**, Amer. Math. Soc., Providence, RI, 628pp.

T. Leistner, P. Nurowski, K. Sagerschnig, *New relations between $G_2$-geometries in dimensions $5$ and $7$*, Internat. J. Math. **28**(13) (2017). arXiv:1601.03979