The appropriate and fair thing looks like this.
Every employee gets some bank of sick time, personal time, and vacation time. When the employee is sick, or has to care for a sick dependent or parent, they get to take the time out of that bank. If they run out of time in that bank, they can still take the time off, but it’s unpaid.
This does mean that people with sickly children spend their vacation time caring for the children instead of taking two weeks and going to Disneyworld, but hey, them’s the breaks. They also spend more on antibiotics and bleach, but it’s not reasonable to expect their coworkers to contribute to the cost of those as well.
Also, for other circumstances, there are things like maternity or paternity leave, short-term disability, and leaves of absence. I don’t think it’s ridiculous for a mother to want to spend more than six weeks at home with her newborn, but I do think it’s ridiculous to expect the employer to continue paying her beyond that.
This is close to what my current employer does. The principal difference is that there is no sick time bank, but if you’re out for three days in a row, you need a doctor’s note. And if you take too much sick time, your manager can bring it up with HR, and they will notify you that for the next several months any sick time will come out of vacation time or be unpaid time off, at your option.
And I agree with @JLeslie that employers need to be flexible with everyone. What pisses me off is not that Fred gets to leave early to pick his kid up from daycare; what pisses me off is when a boss lets Fred have flexible hours to take care of his commitments outside of work, but doesn’t let me have flexible hours to take care of my commitments outside of work, when the only material difference is that Fred’s commitments involve kids and mine don’t.