Are there any differences between "supernatural" and "paranormal"?
As super means above and para means beyond, I would say the differences lie in the parts natural and normal.
In the mere word sense something paranormal is just beyond normal. Natural can be interpreted as synonym to normal in the sense of something comes natural and the like. It can be interpreted as another realm as well, the realm of God as opposed to the realm of man.
As @Aidas would use paranormal for the mysterious, one could argue that the realm of God is paranormal. But it is very unusual to talk of the realm and the workings of God as paranormal, while it is perfectly normal to call effects believed to be caused by UFO's, aliens (from outer space), or ghosts paranormal.
In addition it seems more fitting to call effects paranormal, while supernatural would also be used for beings.
paranormal is translated into German as übersinnlich, above the sense realm, while supernatural is translated überirdisch, übermenschlich, übernatürlich: above earthly, above humane, supernatural.
Being not a native English speaker, I would use "paranormal" for mysterious and unexplainable, whereas I would use "supernatural" for fictional and superior than normal.