I'm surprised to have to explain this to an adult, but well here we are.
All sources of information are not created equal. It is necessary to think critically about the source of your information, when evaluating its weight - not just blindly accept everything you read on the internet as fact.
Workingfamilies.org is a pressure group - it has a political agenda to effect change to legislation to promote the interests of working families, and the things it publishes on its website do not have the force of law. So, it is likely to present information in a manner that advances this agenda. It has a vested interest in highlighting that parenthood is not a protected characteristic in the Equality Act - because it would like it to be so and indeed campaigns for this change to EqA. It has no incentive to then go on to set out that actual legal position, which is that the courts have read into the act that discrimination against people with childcare responsibilities amounts to discrimination on the grounds of sex, even though it is not defined as a protected characteristic.
On the other hand, judgments handed down by The Honourable Mrs Justice Eady DBE in the Employment Appeal Tribunal very much are binding.
As I have said several times now, the law of England and Wales isn't just made up of statutes. The common law also derives from case law, i.e. what courts have decided in the past (the doctrine of stare decisis).
So, when the courts have consistently found that the care responsibilities for small children disproportionally impact women,it becomes part of the law - Judge made law is definitely a thing. So, discriminating against employees with such responsibilities also becomes unlawful. They even coined a term for it - the Childcare Disparity.
I did provide a link above with a helpful analyis of the evolution of this point of law, the option to actually read it remains available to you.
Edited to add - if a disabled person needs changes to their contract as a reasonable adjustment, then the employer must fairly consider the request and make the adjustments if it can. I'm not sure what point you're trying to make here.