Nah. Maybe sometimes I lean forward and back a little to judge something, but people with binocular vision do that too.

Most of our depth perception isn't from stereopsis or other binocular cues.

Swiveling doesn't reall give parallax, you need lateral linear motion, think of a soccer goalie getting into position.

Rotating your head will give some lateral motion (your eye moves in space since the centre of rotation is your neck). You can see this easily by turning your head while focusing on an object in the foreground, your perspective on it will shift.

Side to side motion is actually what I meant and what I do sometimes in poor weather driving or when I have branches up close and blocking my vision while trying to see beyond them.