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