Zariski-dense surface groups in non-uniform lattices of split real Lie groups

For $\textrm{SL}(n,\mathbb{R})$ ($n\geq3$), $\textrm{SO}(n+1,n)$ ($n\geq2$), $\textrm{Sp}(2n,\mathbb{R})$ ($n\geq2$) and for the adjoint real split form of the exceptional group $\textrm{G}_2$, we exhibit non-uniform lattices in which we construct thin Hitchin representations by arithmetic methods. These representations give infinitely many orbits under the action of the mapping class group (except maybe for $\textrm{G}_2$). In particular, we show that when $p\neq2$ is prime every non-uniform lattice of $\mathrm{SL}(p,\mathbb{R})$ contains thin Hitchin representations.