The phrase "that can't be explained by science" is meaningless. That's kind of like saying "that can't be explained by explanations." If we are using "Paranormal" with that meaningless definition, then there is no such thing as a paranormal occurrence, as everything can be (eventually) explained.
In the real world, we don't have explanations for lots of things. Not just spooky houses and UFOs, but loads of mathematical equations, dark matter/energy stuff, cosmological constants, the origins of everything. Are these paranormal phenomena? No, they're just unexplained. Not unexplainable, but unexplained. Even if the answer is God and Jesus and Santa Claus, that's still an explanation that science has yet to identify (and may never identify).
In Star Trek Land, the unexplained is everywhere. And fantastical explanations (higher beings like Q and Organians, amazingly advanced technologies like warp travel and transporters connected to fantastical realms like subspace and the mycelial network) abound. Ghosts exist, in the form of crazy aliens, but still ghosts in every conceivable definition. People have come back from the dead, due to super-science or crazy science, but that's still resurrection. The Gods are real (Apollo, Kukulkan, Lucifer), even if the stories we came up with didn't quite match their background.
And magic is a thing that people can do. But only in Megas-Tu. Although it bears a strong similarity to the thought powers displayed by Wesley Crusher and the Traveler.