[Yoda]

Is there any place in the back swing or forward swing where the hands move parallel with the ground?

The Hands are responsible for producing and maintaining the Three-Dimensional, On Plane orbit of the Clubhead throughout the Golf Stroke:

-- Up-In-and-Back from Address to the Top of the Backstroke;

-- Down-Out-and-Forward From the Top to the Low Point the Downstroke;

-- Up-In-and-Forward from Low Point to the Finish.

This Three Dimensional Motion of the Clubhead requires that the Hands describe a similar On Plane orbit. Accordingly, the Motion of the Hands during the Stroke is oriented parallel to the Clubhead's Delivery Line (usually the Plane Line). However, holding the Clubhead (and the Hands) to a level or "parallel to the ground" Path is Steering, (the First Snare / 3-F-7-A). Hence, unless such a Path is employed intentionally to produce a special effect, it should be avoided.