Here is a proof which relies on a straighforward generalization of the Ping Pong Lemma.
Claim. Let $a$ and $b$ be the transformations of the Riemann sphere $\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$ defined by $$ a(z) = \frac{1}{z} + 2,\quad b(z) = z + 4. $$ Let $d \in \mathbb{N}_{> 0} \cup \{\infty\}$. Then the subgroup of $\operatorname{PSL}(2, \mathbb{R})$ generated by $X_d = \{b^{k}ab^{-k} \,\vert\, 0 \le k < d\}$ is free on $X_d$.
The claim and its proof are a trivial adaptation of [1, Example II.B.26] which makes use of Schottky groups.
Proof of the Claim. Let $\mathbb{D} = \{ z \in \mathbb{C} \, \vert \, \vert z \vert \le 1\}$, $\Omega_k = (\mathbb{D} + 4k) \cup (\mathbb{D} + 4k + 2)$ for $0 \le k < d$. Observe that $\gamma_k = b^ka b^{-k}$ maps
- the exterior of the disk $\mathbb{D} + 4k$ onto the interior of the disk $\mathbb{D} + 4k + 2$, and
- the exterior of the disk $\mathbb{D} + 4k + 2$ onto the interior of the disk $\mathbb{D} + 4k$.
Thus $\gamma_k^m (\Omega_l) \subseteq \Omega_k$ for every $k \neq l$ and every $m \in \mathbb{Z} \setminus \{0\}$. Clearly $\Omega_k \nsubseteq \Omega_l$ if $k \neq l$. A straightforward generalization of the Table-Tennis Lemma yields the result.
[1] P. de la Harpe, "Topics in Geometric Group Theory", 2000.