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
Eisenhart, L. P., Riemannian geometry, Princeton: Princeton University Press. vii, 306 p. (1949). ZBL0041.29403. 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.