I believe the law is that they have to give you time off but they don’t have to pay you, so unpaid leave is all you’re legally entitled to
Family leave is unpaid (very rear occasions you might get few days paid leave), so your employer is right by saying best to keep some annual leave allowance for this unless you can afford to have many days unpaid.
Completely depends on who you work for, they have to give you the leave but don’t have to pay it. They can’t make you use holidays for it either. Mine is classed as emergency paid leave which I’m very grateful for!
Unpaid parental leave
You are entitled to dependency days (Google them). They are typically unpaid unless you have a very generous employer. You could use AL if you'd like and then you'd be paid for it I suppose but the dependency days are there too if you don't have any AL left or you don't want to use it. Also that's very presumptuous of your employer 😂😂 I've been back at work since last May and I've not had to take any time off due to child sickness.