The best way to get an idea of what parapsychology is to talk to parapsychologists, read academic research, and to attempt to get work experience with a parapsychologist. I would be cautious of approaching people who refer to themselves as "parapsychologists", but who have no academic credentials to substantiate this claim. This can happen quite a lot on the internet.
Parapsychology is only a very small field with a handful of research centres in the United Kingdom. It is my impression that currently the U.K. probably has a greater representation of parapsychologist than anywhere else in the world. Details of the main U.K. universities and other institutes engaged in parapsychological research can be found at the Koestler Parapsychology Unit at the University of Edinburgh. There is also a well-known private research institute in Germany called the Institut für Grenzgebiete der Psychologie und Psychohygiene, as well as the Rhine Research Center in the USA.
The biggest issue that faces parapsychology (and indeed all research) at all times is funding. Regrettably there are very few organisations which fund parapsychological research and this obviously restricts the chances that one has of finding a permanent position or funding for postgraduate study. However, an advantage of parapsychology is that is draws people from many different disciplines and backgrounds, so it is possible to enter the field with a PhD in a different area, or do the occasional project separate to a primary discipline.
I would suggest any student who is serious about getting a feel for what parapsychology is all about to get a copy of Irwin and Watt's (2007) "An Introduction to Parapsychology". Now in its fifth edition, it is the set-text for several undergraduate parapsychology modules that I am familiar with. Although it is a bit expensive and doesn't cover all of the topics that I would like it to, I think that it should give you a pretty good introduction to what parapsychology is. I must warn you, it is pretty dry and academic, but that's what many of us are!
Basically, it is not possible to get a degree in parapsychology. Many parapsychologists get their first degree (a Bachelor of Science, or B.Sc) in psychology, sometimes with elements of parapsychology in it, but primarily in a wide variety of subjects in psychology. This is then often followed by an Masters of Science (or M.Sc) in psychology, which is then followed by a Doctorate in Psychology (or a Ph.D). It is possible to study more and more elements of parapsychology as you go through these different degrees. This whole process can take many years, I'm afraid. I will try and summarise the process as best I can, beginning with pre-university study (as some students reading this may be at this stage).
Read my guide on "Becoming a Parapsychologist". I wrote this to try and advise some of the students who contact me asking about this complex issue.