Let $X$ be a $\delta$-hyperbolic geodesic space. Then we have the following classification of isometries on $X$:
Theorem: Let $g$ be an isometry on $X$. Then, exactely one of the following case holds: (i) $g$ is elliptic: for every $x \in X$, $\{ g^n \cdot x : n \in \mathbb{Z} \}$ is bounded; (ii) $g$ is parabolic: the action induced by $g$ on the boundary $\partial X$ has exactly one fixed point; (iii) $g$ is hyperbolic: the action induced by $g$ on the boundary $\partial X$ has exactly two fixed point.
Now let $G$ be a non elementary (ie. neither finite nor virtually $\mathbb{Z}$) hyperbolic group. If $X$ is a Cayley graph of $G$ then $G$ has a natural isometric action on $X$ and:
Theorem: $G$ has no parabolic isometry.
I would like to generelize this theorem for other hyperbolic space $X$. For example, let $G$ be a non elementary hyperbolic group acting freely on a hyperbolic geodesic space $X$ by isometry; can $G$ contain a parabolic isometry? (Feel free to modify the hypotheses.)
Edit 1: We can add the following requirement (where $X$ is $\delta$-hyperbolic): for all $x \in X$, $B(x,n_0\delta) \cap G \cdot x$ contains at most $C_0$ points, where $n_0$ and $C_0$ doesn't depend on $x$.
Yes, there can be parabolic isometries in your sense. Examples can be obtained by "cone constructions", e.g. as follows:
Let $\mathbb{H}^2$ be the upper half-plane and let $X = (0,\infty) \times \mathbb{H}^2$ with the Riemannian metric $(dt)^2 + e^{-2t}\frac{\lvert dz\rvert^2}{y^2}$, which is a $\operatorname{CAT}(-1)$-space and hence geodesic and $\delta$-hyperbolic. Then $\operatorname{PSL}(2,\mathbb{R})$ acts freely by isometries on $X$ while it fixes a point in the boundary.
Any non-elementary Gromov-hyperbolic subgroup of $\operatorname{SL}(2,\mathbb{R})$ will yield an example.
A rather silly sufficient condition is to assume properness and cocompactness of the action, because the Svarc-Milnor lemma reduces this (up to quasi-isometric equivalence) to the case you already know.