Paranormal research is by name a kind of research. But this does not imply it is scientific research that falls under any branch of science.
In fact, by definition it is not scientific because "paranormal" is something that "cannot" be explained by science. Science is the effort to explain things that "haven't" been explained yet.
Though, in principle, you could have branches of science studying "paranormal beliefs". For example, Psychology can study why a person holds such beliefs and how that affects its phyche: usually conspiracy-theorists and "paranormal researchers" are feeling that they hold a special place due to the fact they're "out of the norm." Sociology can study how these mechanisms manifest to groups of people, or History can record such behaviors.
Any definition of modern science and the practice of scientific method, rule out the notion that paranormal research can give testable scientific theories supporting that there are "paranormal" phenomena.