Some thoughts and a brief answer – maybe not a lot.
You already have the right to apply for permission to make section 8 applications ie for contact, residence, specific issue orders or prohibited steps orders. You can apply in care proceedings to become a party, and this is almost never refused.
The court is very unlikely to deny you that permission if you have a genuine concern about the way your child is being treated.
For example, if a mother with care is denying you information about or contact to your child you can make applications as above.
A mother may feel that she cannot have a reasonable conversation with you because of the circumstances surrounding your separation. She may, rightly or wrongly, see you as an irresponsible father.
The mother may simply not understand what sharing parental responsibility will mean on a day-to-day basis and may need to take legal advice.
Sometimes the issue of parental responsibility is seen as a bit of a footnote to other major decision such as where should the child live. It may be very difficult for the mother to consider the question of parental responsibility before the main issue has been decided.
The mother may need to hear from you that her role in the child’s life is not basically challenged – for example, that you agree the child should live with her and that you acknowledge that she is a good mother.
I am not suggesting that you should give up on parental responsibility but you might want to think through whether and maybe more importantly, when to pursue the application.
When a relationship has gone wrong, it takes time to restore the trust. It may be that the issue of PR can wait and other things such as contact are more important. Is there some way of introducing a third party as mediator or medium for communication? Your mother – her mother – a sister etc.
You may have a great deal to contribute to your child’s future. The mother may unreasonably be refusing to agree that you should have parental responsibility. If that is the case then maybe you need to go for it to make the point about your status in the child’s life.