Reckon they're under the power of their parent, perhaps? (9,6)

'under the power of their parent perhaps?' is the definition.
I don't know anything about this answer so I cannot tell whether it can be defined by this definition.

'reckon' is the wordplay.
I cannot really understand how this works, but
'reckon' could be 'figure' (synonyms) and 'figure' is found within the answer.
The remaining letters 'authority' is a valid word which might be clued in a way I don't see.
This may be the basis of the clue (or it may be nonsense).

'they're' is the link.

Can you help me to learn more?