Correction:
Today, I have done this experiment with a larger magnet whose magnetic field has much smaller vertical component. But the wire did not move. When I approached the wire to the edge of the magnet where the vertical component is stronger, the wire moved. So, the yesterday's conclusion was wrong: the movement of the wire was due to the the vertical component near the edge but not due to force parallel to current.
I apologize for having posted a wrong result.
However, I think it's useful to keep this post for those who have already read it in order to inform them this correction. It is also good for me to remember this disaster.
Sorry again.
PengKuan