UK Born child to parents with Refugee Status - 5 Years

Hello

I'm new to the forum so please excuse me if this is not the correct section.

What does one need to do if they currently have Refugee status with 5 years' leave and they have a child in the UK?

The birth has been registered and birth certificate obtained. Does the parent need to register the birth with the Home Office? What do they need to apply for in order to obtain legal residence in the UK?

From what I've read (Googling), it sounds like the child will be granted the same leave in the UK as the parent so 5 years, if this is correct, what form should one use to apply for this and what form should one use to apply for some sort of travel document for the child should the parents travel?

Thank you in advance