Well, in the field of motion graphics (which openings like this fall under), what’s happening is there’s a bunch of 2D imagery (either stills or video/motion), that are mapped onto planes and manipulated in Z-space (X = the left & right dimension; Y = the up & down dimension; and Z = back and forth) using motion graphic software like After Effects, Motion, Nuke, etc.
There’s some overlap into true “3D/CGI” animation here though (not to be confused with stereoscopic 3D). A lot of these sorts of effects can be pulled off in traditional 3D packages such as Maya, 3D Studio Max, Cinema 4D and so on. But it usually all comes together in a motion graphics package like After Effects, because while its roots were focused on animating 2D elements, these software packages now include true Z-space, allowing the motion artist to do some very interesting, dimensional effects otherwise very difficult to pull off in days of yore.
So, to say all that using one term, it seems 2.5-D is now the broadly accepted shorthand term for this technique.
Parallax is a much more specific term in CG denoting the distortion of angles you get from using different virtual lenses and fields of view. A strong parallax would indicate your lens is very wide, giving you exaggerated angles and vanishing points. A weak parallax would indicate a more telephoto appearance, with the relative distances appearing crushed and flat.