Fitting UI element height to content while expanding width to fill screen?

This may help you:

• Create a Canvas

• Set the Canvas Scaler UI Scale Mode to Scale with Screen Size

• Add a child Game Object to the Canvas

• Set the Content Size Fitter Horizontal to Unconstrained

• Set the Content Size Fitter Vertical to Preferred Size

• Set the child Rect Transform Min X anchor to 0 and the Max X anchor to 1

• Set the child Rect Transform Left and Right to 0

• Add both a Vertical Layout Group and a Content Size Fitter to the child Game Object

• Set the Vertical Layout Group Spacing to 4,

• Set the Vertical Layout Group Child Controls Size Width and Height to true

• Set the Vertical Layout Group Child Force Expand Width and Height to false

• Add a button (or any object really) as a child of the previous game Object

• Add a Layout Element to the newly created button

• Set the newly created button's Layout Element Preferred Width and Height to... (e.g. 160 and 30)

• Duplicate the button

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I hope that may help you!

Place all your button in a container (it might be the parent object you mentioned).
Make your container with anchor min (0, 0) and max (1, 1). Set the Left. Top, Right and Bottom to 0. Now your parent box occupies the whole screen. You can also click the square icon above Anchor and then press Alt and press the bottom right icon. That will set it all to what I just described.
If you have a limited and constant amount of button, do not use a vertical layout group, it is convenient but adds computation for redrawing that you can avoid. Make them all children of the parent container object. Select them all four and set anchor min X to 0 and max X to 1. Then set left and right to 0 for all four of them.

Now comes the tricky part, for each from top to bottom set the anchors to:

1. minY : 0.75 maxY : 1

2. minY : 0.5 maxY : 0.75

3. minY : 0.25 maxY : 0.5

4. minY : 0 maxY : 0.25

If you end up with a weird amount of button like 7 of them, you can just put 1/7 then 2/7 then 3/7 and so on, unity will set the value for you. Once you set the all fine, select them all and set Top and Bottom values to 0.

This is it, they should occupy the whole width and each will use a quarter of the height. It will also rescale based on the screen dimensions, you can even change the ratio from landscape to portrait.

And it takes no additional component or code.