OP has a point. In a usual situation, the parent whose career/ pay will be affected less by a day off, will take the day off for sick day/ snow day whatever.
On the whole, that would be the woman.
^ I don't think anything I've said so far is offensive.
Yes we can blame it on our patriarchal society, but whatever the reason, this is usually the case.
However, I said "on the whole" and not "always".
I took a day off before Xmas with a sick DD. My DP and I had discussed this and as my job was the better one with more opportunity for progression, we had previously decided that he would be responsible for sick days etc.
However, my DP was self employed at the time and I work in the public sector where I would not be docked pay or anything, and so I took the time off.
I did feel extremely guilty about using the excuse that 'DD is ill' and did feel that if I was a man then my female colleagues would have probably been discussing what a wonderful family man I was.
I think I'm rambling now
but what I mean is, the OP made a decision that is best for his family, as did I. The sad situation is that for most families, the woman's career is the one put on hold for the children. This leads to it making more sense that the woman is the default carer. Now I'm not agreeing with this but this is how it works in most situations (not mine).
Maybe when man can carry and give birth to a child, things might be more equal! 