Here's the legal position (am employment solicitor):
You are allowed a very limited amount of (unpaid) time off to make arrangements to have a sick child cared for, but you are not entitled as of right to time off, paid or unpaid, to look after the child until it's better. This probably translates as about 1 day.
If you take off an unreasonable amount of time even on this basis, you could be disciplined and eventually dismissed with notice. There might be some leeway due to a parent with a disabled child, but that's about it.
The law expects employers to bear with employees in a real emergency, but employers are entitled to expect employees to have contingency plans in place so that they are not absent each time a dependent relative is ill.
If however there is evidence that you are being censured for this type of absence more than other employees who need time off, this might be discriminatory.
I act for both employers and employees, so see both sides of this problem. Whilst it does not make life easy for employees with children, arguably it is not reasonable to expect employers (and other employees) to suffer regular or protracted disruption to their work to accommodate family arrangements. If for example a couple co parent a child and the employer of one parent enforces a strict policy about this, the less strict employer of the other parent will inevitably end up suffering the disruption.
Employers are also advised to resist relaxing the rule for one employee because this creates a risk of creating a cause of action of another employee is treated less favourably.
I realise this is a bit of a "dry" answer, but I hope it helps.