Final answer:
When a star's motion is parallel to Earth and it is neither moving towards nor away from us, there will be no Doppler shift observed in its spectrum. The spectral lines would appear at their rest positions, indicating no radial velocity relative to Earth.
Step-by-step explanation:
When stars orbit each other in a system where their motion includes components parallel to Earth, we are able to observe shifts in their spectra due to the Doppler effect. If a star moves toward us, its spectral lines shift toward the blue end of the spectrum, and if it moves away from us, they shift toward the red end. When the stars' motion is perpendicular to our line of sight, their radial velocities equal that of their system's center of mass, resulting in their spectral lines appearing superimposed, thus, indicating no blueshift or redshift.
In the case where a star is parallel to Earth and not moving toward or away from us, we would not observe a Doppler shift in its spectrum. The observed spectral lines would remain at their rest positions without any blueshift or redshift, indicating no radial velocity relative to our observation point.