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 songs: parallels (even fifthsfifth parallel) are possibleusual creating a typical effekteffect.
In European folk musicFolk Music parallels of thirds and sixths are usual, in English and American folk songsFolk Songs and Pop songsSongs (Beatles) parallel fourth with the tune are also possibleusual.
try this out:
Isn't it weird that even though harmony went down in pitch I sing higher in pitch at the same time?
summary:
in almost all styles it is even better to lead the voice in opposition to the bassline respectively the chords.