There’s no such thing as parental rights. You have a responsibility to your child, not a right over them.
Parental responsibility is granted in all cases where there’s no dispute over paternity, and there isn’t a hugely harmful reason why not. I’m a childcare barrister and have been for 10+ years. I can think of two cases where PR was refused, and both were gang members who had put out orders to kill the mum and child.
Having PR (or not!) has nothing to do with maintenance. The CMS is entirely separate. Parents without PR still have to pay maintenance.
Ultimately, even if he has PR there’s nothing compelling him to use it, so there’s really no downside to it from his perspective. I can’t think what would put him off.
The purpose of PR isn’t really for the parents - it’s for the child, for his or her identity and to show that both parents have a duty of responsibility towards them, whether they choose to use it or not.
To come back to your original question - a solicitor should advise on the likely prospects of success, any benefits and risks, the costs of pursuing an application and so forth. If the client instructs them to proceed against advice, they still have to do so.