Firstly, stop calling it 'supervised visits' and start referring to 'supported contact'.
Secondly, he's never going to be able to build up a relationship by going to a park.
Offer him a structured increase of time, put it in writing. Every other day for a couple of hours, at your home, or at a family member's home (his family if he has someone you trust), and as your child starts to recognise him and seek him out for comfort, or can take comfort from him, then let him take her out for a walk or to the park for half an hour, increasing it once you have seen he can manage.
Show him how to make up food for her (at 10 months old, she should be eating), what she eats, what she likes, how to make up bottles of water, or anything else she might need whilst he is out with her. Teach him how to change nappies, how to notice if she is distressed etc.
There is literally no point in you attempting to fight this in any way, and the more on board you are with it, the more you help the father to understand the child's needs, the easier it will be for you to let him look after him. That is in the child's best interests, as long as there are no safeguarding concerns.
He will do one of two things, he will either step up and accept the help you can give him and build up to having the child alone, or he will fight against it and drag you into court. At which point both of you are giving up the right to make decisions about your own child, and handing it over to third parties who don't know her to decide what is best. Think about that.
He has to at least attend a mediation information session before applying to court. You should engage if a mediator contacts you - better to agree a parenting plan between you with the help of a mediator than go to court.