You can constrain an individual axis with the Aim constraint. For example you can have the child object only affected on its Y-axis.

To constrain an axis with the Aim constraint:

1. Double-click the Aim constraint in the Scene browser to select it.
2. Open the Properties window. The Aim constraint settings display.
3. Expand Constraint Axes and disable the X, Y, or Z axis for the desired translation effect you want to inhibit.
4. To reactivate the unconstrained axes, activate the option.