Are two lines that lie in parallel planes always parallel?

Of course not.

-- Take a washable marker, and give your little sister another washable marker.

-- Stand back-to-back in your bedroom, you facing the front wall, and her facing the back wall.

-- When you say 'GO', both of you draw a straight line on the wall.
   Then look at the lines.

-- The front and back walls are parallel planes, but it would be
amazing if the two lines are parallel.  One could even be a
horizontal line and the other a vertical one . . . definitely not parallel.