I'm not sure if you want follow the bass line or the back voices or invent a melody:
It depends of the style:
In classic, Jazz and Pop the you may keep on this rules:
bass: you can choose the pitch you like - most comfortable to your voice (priority: the counter-point to the tune).
back-voicing: keep the common tones or go to the next step up or down (preferably counter movement to the melody or bass line.
In Gospel parallels (even fifths are possible creating a typical effekt.
In European folk music parallels of thirds and sixths are usual, in English and American folk songs and Pop songs parallel fourth with the tune are also possible.