Using any of the models like the Poincare open disk model, this turns into a simple Euclidean geometry problem.
I have some experience in hyperbolic geometry but unsure in proving the following:
Let u and v be parallel lines, and let Q be on a point on u and R be a point on v. Prove that there exists a line w such that Q and R are on the opposite sides of w, and w is parallel to both u and v.
Trying to prove this without the Poincare open disk model. I've made some advancement. I know there are two cases in this proof. Case 1 being the u and v admit a common perpendicular, and Case 2 being u and v are asymptotically parallel. I can prove Case 1 with ease. I'm not sure how to go about constructing w in Case 2.
If u and v are asymptotically parallel, then they define a unique "point at infinity" ("ideal point"), m, where they intersect. Take p and q to be any two points on u and v. Then there exist a unique line, l, through p and q and so a point, r, between p and q on that line. The line defined by r and m is parallel to both u and v.