The Galaxy's stellar halo seems to be a tangle of disrupted systems that have been tidally stretched out into streams. Each stream approximately delineates an orbit in the Galactic force-field. In the first paper in this series, we showed that all six phase-space coordinates of each point on an orbit can be reconstructed from the orbit's path across the sky and measurements of the line-of-sight velocity along the orbit. In this paper, we complement this finding by showing that the orbit can also be reconstructed if we know proper motions along the orbit rather than the radial velocities. We also show that accurate proper motions of stream stars would enable distances to be determined to points on the stream that are independent of any assumption about the Galaxy's gravitational potential. Such ‘Galactic parallaxes’ would be as fundamental as conventional trigonometric parallaxes, but measurable to distances ∼70 times further.