If you rotate your camera around some randomly chosen point, your images will have parallax and your stitches will be bad.
With most lenses, there is one special point around which you can rotate your camera and get no parallax.
In the panorama photography community, that special no-parallax point is called the "nodal point" or "entrance pupil".
Periodically there is debate about what it ought to be called. In the physics and optics communities, the term "nodal point" means something else entirely, a place in the optical path that is of no significance to most photographers. Those communities have no special name for the no-parallax point, and their term "entrance pupil" properly refers to a region around the no-parallax point, rather than the point itself. (The entrance pupil is the image of the limiting aperture or diaphragm, as seen through the front of the lens.) Sigh...
Everything considered, the best term for the no-parallax point is probably "entrance pupil". But if you do say "nodal point" while talking about panoramas, everyone will understand what you mean.
Regardless of what you call the no-parallax point, it is easily found by trial and error. Just adjust the rotation point so that foreground and background points stay lined up.
Some lenses, notably fisheyes, do not have a single no-parallax point because the location of the entrance pupil depends on the angle away from the lens axis. With such lenses, it is good to pick one angle at which you like to stitch, and rotate your camera around a point that gives no parallax at that angle.
To facilitate finding the no-parallax point for other people, please fill the measurements you have found for your Camera / Lens / Focal Length combination in the Entrance Pupil Database
Big Ben's Panorama Tutorials, Determining the Nodal Point of a Lens.
Alain Hamblenne, The Grid