Answer
A torque is exerted by one on the other. See answer.
Work Step by Step
Let the two wires be horizontal. Suppose the lower one, wire 1, carries a current pointing east (positive x-direction) and the upper one, wire 2, carries a current pointing north (positive y-direction).
The magnetic field from wire 1 points upward (toward the sky) on the northern half of the upper wire, and points downward (into the Earth) on the southern half of the upper wire.
Use the right-hand rule on the northern half of the upper wire: the force points to the east. Use the right-hand rule on the southern half of the upper wire: the force points to the west. There is no net force, but the upper wire experiences a clockwise torque (into the Earth) about a vertical axis due to the magnetic forces from the lower wire. This torque tends to align the two wires to make their currents parallel.
In the same fashion, one may check that the lower wire experiences a counterclockwise torque, which also tries to align the 2 currents.
