Let $g \ge 2$ and $\Gamma := \langle a_1, b_1, \ldots, a_g, b_g\ |\ [a_1, b_1] \cdots [a_g, b_g] \rangle$, and let $\rho : \Gamma \to \mathrm{PSL}(2, \mathbb{C})$ denote a representation which is discrete and faithful. Can $\rho(\Gamma)$ contain a parabolic element $\rho(\gamma_0)$?
I know that $\rho(\Gamma)$ cannot contain a parabolic element if one replaces $\mathrm{PSL}(2, \mathbb{C})$ by $\mathrm{PSL}(2, \mathbb{R})$. Furthermore, Claim 2.28 of [GGKW] implies the sequence $\{d_{G/K}(K, \rho(\gamma_0)^n K)\}$ is $O(\log n)$, where $G := \mathrm{PSL}(2, \mathbb{C})$, $K := \mathrm{PSU}(2)$, and $G/K \cong \mathbb{H}^3$ denotes the Riemannian symmetric space of $G$ (which identifies with the upper half space model of hyperbolic $3$-space). So $\rho$ must not be a quasi-isometric embedding (with respect to the word metric on $\Gamma$), which, according to Remark 2.36, equivalently means $\rho(\Gamma) \subset \mathrm{PSL}(2, \mathbb{C})$ is not a convex cocompact subgroup. (The notes by [K] have a proof of the equivalence, in Lemma 2.3.)
I believe one kind of example of a non-convex-cocompact surface group embedding arises as a holonomy representation of the fundamental group of a surface fiber in certain mapping tori; specifically, those mapping-tori $M_\phi$ arising from a pseudo-Anosov homeomorphism $\phi : \Sigma \to \Sigma$ of a higher genus oriented closed surface. It follows from the mapping torus case of the hyperbolization theorem that $M_\phi$ admits a discrete, faithful, and cocompact representation $\mathrm{hol} : \pi_1(M_\phi) \to \mathrm{PSL}(2, \mathbb{C})$; see [O]. Since $\pi_1(\Sigma)$ is a non-elementary normal subgroup of $\pi_1(M_\phi) \cong \pi_1(\Sigma) \rtimes \mathbb{Z}$, the limit set of $\mathrm{hol}(\pi_1(\Sigma))$ is all of $\partial \mathbb{H}^3 = \widehat{\mathbb{C}}$, but $\mathrm{hol}(\pi_1(\Sigma)) \backslash \mathbb{H}^3$ has infinite volume (as it contains $\mathbb{Z}$-many copies of $\mathrm{hol}(\pi_1(M_\phi)) \backslash \mathbb{H}^3 \approx M_\phi$). This kind of example doesn't directly answer my question however because $\mathrm{hol}(\pi_1(\Sigma))$ is a subgroup of a uniform lattice of $\mathrm{PSL}(2, \mathbb{C})$, and so is purely hyperbolic.
[GGKW] Guéritaud, François; Guichard, Olivier; Kassel, Fanny; Wienhard, Anna, Anosov representations and proper actions, Geom. Topol. 21, No. 1, 485-584 (2017). ZBL1373.37095.
[K] Kassel, Fanny; Convex Cocompact Groups in Real Rank One and Higher, Notes for a talk at the GEAR workshop on Higher Teichmüller–Thurston Theory, Northport, Maine, 23–30 June 2013.
[O] Otal, Jean-Pierre, The hyperbolization theorem for fibered 3-manifolds. Transl. from the French by Leslie D. Kay, SMF/AMS Texts and Monographs. 7. Providence, RI: American Mathematical Society (AMS). xiv, 126 p. (2001). ZBL0986.57001.
There are several ways to construct such examples. One way is to use a Maskit Combination theorem (for amalgamated free product), amalgamating two single-cusped Fuchsian subgroups along a parabolic subgroup. You can find such constructions in the books B.Maskit "Kleinian groups" or/and Krushkal, Apanasov, Gusevskii "Kleinian groups and uniformization in examples and problems."
A better (in a sense) way is as follows. Start with a genus $g$ Fuchsian group $F$. Let $U^\pm$ denote the disjoint $F$-invariant open disks in $S^2$. Then the space of quasiconformal deformations of $F$ can be parameterized as $T(U^+/F)\times T(U^-/F)$, where $T$ denoted the Teichmuller space. Now, pick a simple essential loop $c$ on $U^+/F$ and let $D_c$ denote the Dehn twist along $c$. Fix $(\tau, \tau)\in T(U^+/F)\times T(U^-/F)$ and consider the sequence of quasifuchsian representations $\rho_n: F\to PSL(2,C)$ corresponding to the sequence of pairs $(\tau, D_c^n\tau)$. One can show that $\rho_n$ subconverges (and even converges) to a representation $\rho: F\to PSL(2,C)$ such that $\rho(c)$ is parabolic. The representation $\rho$ will be necessarily discrete and faithful. (I am suppressing the "normalization" issue of the representations since they are only defined up to conjugation in $PSL(2,C)$.) An alternative is to take a sequence parameterizd by $(\tau, \tau_n)$ where each $\tau_n$ is obtained from $\tau$ by fixing all Fenchel-Nielsen parameters except for the length parameter corresponding to $c$, which you set to converge to zero as $n\to\infty$. See for instance Theorem 4.69 and Lemma 4.68 in K.Ohshika's book "Kleinian groups".
More generally, one has the following:
Suppose that $M$ is a compact, oriented, connected, irreducible, atoroidal 3-dimensional manifold with nonempty boundary such that (for simplicity) the boundary contains no tori. Let $c$ be a compact 1-dimensional submanifold of $\partial M$ such that no two distinct components of $c$ are freely homotopic in $M$. Then there exists a discrete and faithful representation $\rho: \pi_1(M)\to PSL(2,C)$ which sends each component of $c$ to a parabolic element. This is an application of Thurston's hyperbolization theorem.