Use expressions, not parenting.
Alt-click the position stopwatch for your soon-to-be-a-child layer. Type an open bracket ([) then pick whip to the x part of the position property of your soon-to-be-parent layer. Then type a comma, a zero, and a close bracket (]) then a plus sign and the word "value". Now, you can adjust the x and y of your "child" layer as much as you would like, but any time the "parent" moves in x, the child will move the same amount.
The expression will look something like this: [pickwhip, 0] + value