The Curve Selection Lemma tells us if $S$ is a semi-algebraic set containing points arbitrarily close to the origin, that is, $0 \in \overline{S}$ then there is a real analytic curve $\alpha: [0, \epsilon) \to \mathbb{R}^n $ with $\alpha(0) = 0$ and $\alpha(t)\in S$ for all $t>0$.
I would like to know if there is an analogue of the curve selection lemma that guarantees the existence of a parameterized surface, say, $$ F:[0,\epsilon)\times [0,\epsilon)\times\cdots\times [0,\epsilon)\to \mathbb{R}^n. $$ such that $F(0,0,\ldots,0)=(0,0,\ldots,0)$ and $F(t_1,t_2,\ldots,t_n)\in S$ for all $t_1>0,t_2>0, \ldots, t_n>0$.
Update. A set is semi-algebraic in $\mathbb{R}^n$ if there are polynomials $p_1, p_2,\ldots, p_\ell$ and $q_1, q_2,\ldots, q_r$ such that $$ S=\{ x\in\mathbb{R}^n: p_1(x)=0, p_2(x)=0,\ldots, p_\ell(x)=0; q_1(x)\leq 0, q_2(x)\leq 0,\ldots, q_r(x)\leq 0 \}. $$
I presume your real question is more focused than the one you gave which is trivially true, take your curve $\alpha(t)$ from the curve selection lemma and let $F(t_1,...t_n) = \alpha(t_1+...+t_n)$ or even $F(t_1,...) = \alpha(t_1)$. Perhaps you wanted a surface parameterization $F(t_1,...,t_k)$ which restricts to a smooth embedding if all $t_i>0$? As long as S is at least k dimensional near $0$ you can do this. Resolution of singularities of real algebraic varieties easily gives a resolution of singularities of semialgebraic sets by which you may as well assume S is an analytic manifold with corners on the boundary. Or use analytic triangulation of semialgebraic (or even subanalytic) sets, see papers of Hironaka or Hardt from the 70s from which I presume this follows By the way your definition of semialgebraic sets is incorrect, you should use strict inequalities $q_i<0$ as well.