Can my parents see the apps I download?

Not from ATT. If you use the same Apple ID as your parents or have family sharing enabled, then they can tell from the AppStore app what apps have been downloaded though not necessarily which device downloaded each app. 

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@sandblaster wrote:

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@Rayosun wrote:
No, they cannot see what data is used for. However if a paid app was downloaded and billed to the carrier, they will be able to see the charge on the bill. AT&T does take privacy seriously.

If they are free apps - you're all good!

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Note this was posted in the Apple forum so presuming OP has an iPhone. Apple apps do not bill through the carrier. 

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Yeah I probably should have worded it better and included the Android reference on the paid apps mention. 😛