"Belief," however, should play no role in the exploration of the paranormal. Belief implies that you're accepting an idea on faith; if you believe something, there's no reason to examine it further -- your mind is made up. When people ask me if I believe in ghosts, I say no. Rather, I tell them that there is evidence for ghost and haunting phenomena reported over thousands of years, but as yet we don't really understand what it is or what it represents. We have various theories, but no one knows for sure. But because it certainly is a part of the human experience, it is quite worthy of investigation.
So my answer is: Yes, the paranormal is real in that there are many mysteries in this life -- from ghosts to psychic phenomena to reincarnation and all the rest of it -- that are not understood or presently explained by current scientific models.
The rest of my answer is: You don't need to believe in the paranormal. Instead, you should find out as much about it as you can, read the case studies and examine the evidence. (Be careful, however, there's a lot of crap out there, both on the internet and in book form. Look for reputable authors.) You'll find that exploring the paranormal in this way is a fascinating, mind-opening journey -- an ongoing expedition into the mysteries that surround us and that may actually be clues to the true nature of our existence on this planet.