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.

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.