What is a parent of a British citizen entitled to?

My family and I have FLR visa ( 30 months) that started in march.
Our child is now 10years and has registered for British citizenship.

As the parents what status are we entitled to after or before our visa runs out. And what other advantages comes with having a British child?

For background into the OP's current circumstances, also see these threads.

Relatively little. I believe that the parents can apply to stay in the UK with the child (which I understand is the route that you are already on), but the parents will be on a 10 year FLR(FP) path and I am not sure whether both the parents can apply together on that basis. But wait for others to advise further.

You will also most likely not be entitled to most non-contributory benefits (Zambrano carers are also non-EEA parents of British citizen children).

You may have more rights if you move to another EEA member-state before 31st December 2020 with the British child and acquire rights in that other EEA member-state under EU law.

PS: Just wanted to clarify, has your eldest child received a registration certificate, as you mentioned in an earlier post that she only reaches 10 years of age in August 2020?