We can define the concept of ''direction'' as the equivalence class of parallel lines, but this is a good definition only if the Euclide parallels axiom is assumed, so that there is only one parallel line to a given line from a given point.
Is it possible to define an analogous concept if this axiom is not assumed and we have different ''parallel'' lines to a given line? In other words, the concept of ''direction'' is given only in Euclidean geometry or is it defined also in other geometries?
This is one possible translation of the concept. Depending of the way you view the Euclidean concept, it translates to the hyperbolic world in different ways or not at all.
The direction of an unoriented line (i.e. only up to $180°$, not $360°$) in Euclidean geometry can be associated with the set of parallels in that direction (as you already wrote), or equivalently in projective geometry with the point at infinity where all these parallels meet. Translating this to hyperbolic geometry a direction would be an ideal point (a point on the “boundary” of the model), or equivalently the set of limit-parallel geodesics incident with that.
Actually you can define a natural orientation on these in hyperbolic geometry, towards or away from the ideal point. Note that this same kind of orientation definition doesn't work in Euclidean geometry. But the original question here looks non-oriented.