It is a matter of calculation. Take successive derivatives of the conformal Killing equation, until you notice that you can solve for all derivatives of $X$ at a certain order (3rd order here) in terms of lower order ones. I don't know if there is a canonical reference, but a calculation equivalent to such an equation on $X$ is summarized in Eqs. (69.2--5) of

> <cite authors="Eisenhart, L. P.">_Eisenhart, L. P._, [**Riemannian geometry**](https://mathoverflow.net/a/452973), Princeton: Princeton University Press. vii, 306 p. (1949). [ZBL0041.29403](https://zbmath.org/?q=an:0041.29403).</cite> Probably (69.2--5) was already there in the original (1926) edition.

More explicitly, from Eisenhart:
\begin{gather}
\tag{69.1}\label{69.1}
  \nabla_j \xi_i + \nabla_i \xi_j = \psi g_{ij}
\\
\tag{69.2}\label{69.2}
  \nabla_k \nabla_j \xi_i = -\xi_m R^m{}_{kij}
    + \frac{1}{2} (g_{ij} \nabla_k \psi + g_{ik} \nabla_j \psi - g_{jk} \nabla_i \psi)
\\
\tag{69.4}\label{69.4}
  g^{il} \xi_m \nabla_l R^m{}_{kij} - \xi_m \nabla_k R^m{}_j
    - \nabla_k \xi_m R^m{}_j - \nabla_j \xi_m R^m{}_k
    + \frac{(n-2)}{2} \nabla_k \nabla_j \psi
    + \frac{1}{2} g_{jk} \nabla^l \nabla_l \psi = 0
\\
\tag{69.5}\label{69.5}
  \nabla^l \nabla_l \psi = \frac{2}{n-1} (\xi_m \nabla_i R^{mi} + \nabla_i \xi_m R^{mi})
\end{gather}
For completeness, note that basically $\psi = \frac{2}{n} \nabla^i \xi_i$. To get a 3rd order equation for $\xi_i$, start by using \eqref{69.5} to eliminate $\nabla^l \nabla_l \psi$ from \eqref{69.4}, and use the result to solve for $\nabla_j \nabla_i \psi$. Then, take an extra derivative of \eqref{69.2} and eliminate $\nabla_j \nabla_i \psi$.

Feel free to complete the calculation and add the resulting equation to this answer.

----

Probably, this result has been rederived independently multiple times in the literature, or referenced in passing as a "well-known" result without citing any convenient source, as in the article referenced by the OP.

More generally, any PDE on an unknown $X$ which has a differential consequence of the form $\nabla^k X = F(X, \ldots, \nabla^{k-1} X)$ is known as a _PDE of finite type_. While many geometric equations (like variants of the Killing and conformal Killing equations) are known by folklore to be of finite type, it can be surprisingly non-trivial to locate an original or convenient reference for such facts.