OP what do you mean by 'given a really hard time'? What have they said or done?
You are entitled to a day or two unpaid emergency leave, yes. You are not entitled to take the duration of a child's illness off, and you are not entitled to take holiday on such short notice as you have been doing.
Without knowing what 'given a hard time' actually is, it's difficult to say whether your employer is being unreasonable. Also, you say you work part time and have taken 6 days in a month. If you work 2 days a week, that's 3 weeks off at very short notice, or if 3 days a week, that's 2 weeks at short notice.
As squiffy says, what back up childcare arrangements any individual employee has (or has not) made are not the employer's concern.
In reality, as long as time off for a child's illness is split between two parents/both partners and perhaps a combination of annual leave and unpaid emergency leave are used, that would normally be acceptable.
But OP if you've taken maybe 2 or 3 weeks leave at very short notice I can see why your employer is less than sympathetic. Is your son's father around and taking his fair share of time off?