Again, read the guidance I mentioned above. There are requirements for documents to support your application. Although, EEA applications are more relaxed concerning the documents' availability. All you can do is to try your best and submit everything you have. Having both parents' names on the child birth certificate is a very strong evidence of durable relationship.
Although, I would suggest to do more research on case law regarding situations like yours. This was the reason I suggested a legal advice above.
You see, usually there are 2 options exists almost in every endeavor of our life:
1. invest a lot of time in finding a solution to the problem- I mean, when the case is complicated, it is a full time work; you research all possible sources - immigration forms, guidance, forums, groups, websites, case law etc – day in day out, month after month, possibly years
2. find a professional, who knows what he/she does and have an experience in solving the problem. The starting point to find an adviser would be here
https://www.gov.uk/find-an-immigration-adviser. You need Level 2 or 3 OISC. Having an adviser does not mean a 100% success or a quick visa - it still can take months or years, but at least you will free your mind and time.
You can ask some questions on forums or in groups, but bear in mind, that you have to do your homework first and, in order to receive help, ask short questions about particular point you do not understand. Otherwise there will be no answers or the only answer will be to find an adviser. Although, there are immigration advisers on this forum, I doubt they are willing to give you a step by step instructions for free. If you need this type of advice, contact citizens advice here
https://www.citizensadvice.org.uk. It might be a long wait, but they might be able to help you there.
I would go for a DIY option as a last resort – when you visited few advisers and they told you that there is no chance for you to get a visa. It is so easy to talk to up – many people live in limbo after some minor mistakes they did at the beginning.
Therefore, my advice would be: do not try to find a quick fix, do not wait, but start doing something now. Choose from 2 options above and act on it.
As to your illegal status implications - probably, there will be an interview to prove that the relationships are genuine.