Nothing wrong with notifying a new employer that you've taken parental leave - as this leave entitlement is carried over to your next employer it's actually important a new employer knows how much you've taken.
However it sounds as though the problem is your old employer hasn't done that at all. They've basically said you rang in sick for two weeks because your DD was ill. From the new employer's point of view that's really not good, and indeed ringing in as though you are 'sick' when in fact you are looking after a child is potentially a serious concern.
So the first thing to do is correct that, and make sure they realise it's parental leave you took. If they still want to withdraw the offer, then that's obviously a problem. You can't be treated less favourably because you took parental leave.
Get that all straight and see what the new employer says. If all works out fine I wouldn't attempt to take it any further with the old employer tbh, other than requesting a new reference is issued with the correct information.
In terms of details about your illness, no they shouldn't have disclosed that without your express permission. Numbers of days you took off is fine, but the actual nature of your medical condition would be classed as 'Sensitive Information' under the Data Protection Act and shouldn't be shared without your consent.
Contact the new employer, appeal the decision explaining that the reference was inaccurate as it was parental leave you took. I expect they will be happy with that. Contact the old employer and request that a reference is issued for your records at the new employer that is accurate, does not contain sensitive information and gives accurate details of your absence.
See how you get on with that, and come back if you need more help.
For the avoidance of any confusion, it is perfectly legal to provide a 'bad' reference as long as it is factual and can be backed up with evidence. If an employer issuing a reference wants to avoid the risk of legal action from either the individual or the new employer, their reference should not give a misleading overall impression of the individual concerned. That includes giving glowing references about someone when they are clearly not deserved. Unlikely to be any come-back from a new employer who has offered someone a job on the basis of a glowing reference, but if that reference was entirely undeserved, it's a possibility.
It's all a bit of a minefield which is why so many companies now only give basic factual information about dates of employment, absence rates and that kind of thing, and avoid making any kind of subjective comment about people.