I'm an employer and I would say talk to your manager, even if parental leave is unpaid.
From a business perspective what is in a contract and what can be agreed on a case by case basis is often different. Its obvious that you can't work, and obvious that you could ask your doctor to sign you off, so your employer knows full well you will be off and you will need to be paid.
So if they basically know they will need to find a way to work around that and will need to pay you they may offer paid leave or find some other way to make things work,
But as an employer, I would be really annoyed if it felt like a staff member was gaming the system. Even (perhaps especially) if they had been with me for 10 years.
Just an honest conversation saying "I am so stressed, I need to be at the hospital, I can't work while I do it, but I can't afford not to work and my mental health is suffering" could trigger them to arrange something for you that also works out better for them as it might mean you agree to, for example, a regular catch up where you will answer any questions that are confusing whoever is covering your work, which isn't possible at all if you sign off sick.
If they say no, then go to the doctor and get signed off, obviously. But that's fine because you have told them you're stressed and offered them a chance to come up with a solution which works for you.