Does Lorentz force exert a force parallel to the current? NO. However, this experiment shows that, YES, it does exist a force parallel to the current. This experiment makes a plane coil turn in the magnetic field of a magnet. The axle crosses the plane of the coil perpendicularly in the center. Only force parallel to the current can make it turn.

First photography: Will this coil turn?
Second photography: The coil and the magnet.
Third photography: The led will turn on when current flows in the coil. The referential system of the setup.

Please see the video at
http://pengkuanem.blogspot.com/2013/03/lorentz-parallel-action-experiment.html