Not sure what I can say here that hasn't already been said, as all my experiences with women (what few there have been) have ended horribly for me (either by being used and thrown away or taken advantage of for the other party to capitalize on). Double my age and you get a certain Steve Carrell movie.
But anyhow, you said how you felt about the situation in a joking manner, from what I can gather. If you feel she's overexaggerating and it's getting on your nerves, let that be known, but don't come off as downplaying or anything. As it's been said, you both look to one another for support, not to be ridiculed. It may have been a veiled joke to lighten the tension to you, but my rule of thumb with veiled jokes is that if they poke fun at something someone's obviously very stressed over, it's probably best not to say it. Not saying you were in the wrong or anything with trying to lighten the mood, but your secondary response was probably a little out of line. That topic might very well be something that should be talked about, but not when she's in emotional turmoil over it. I don't think what you said was the issue, but moreso how and when you said it. Timing is everything, after all.
I guess my point here would be to support when support is needed, and structure when structure is needed. On both ends. If your significant other's visibly very stressed, I think the best thing to do is just comfort them. It doesn't have to be a clever joke, or overly romantic gesture, but sometimes a simple embrace goes a long way. Any sort of sign that shows you're there for that person while not being over-the-top, and really hugs are just awesome. They can go from friendly to sexual to deeply intimate in a matter of seconds, so anything that versatile is probably a safe bet.
As for how to resolve your current situation, I think apologizing (since the second quip was considerably harsher and probably not very appropriate for that particular time) and acknowledging how she's feeling is a good first step. You don't have to like or even agree with what or how she feels (and by the looks of things I don't think you do), but the bottom line is that she's stressed and could probably use your support right now. That should be your priority over whether or not she's "right" in her feelings. And also to realize that everybody's different and no two people handle things the same way. Yes, there might be that other woman who has to raise a kid and work AND study hard, but assuming that your girl operates the same under a "lesser" workload is actually rather cruel.
I run into this problem a lot, with people asserting that "I don't have it as bad as other people do." Regardless of whether the accusation is "correct" or not (yes, I know I have it better than a half-starved Third-World child, but that doesn't exactly fix whatever my current problem is, and just makes me feel guilty for something I have no control over whatsoever), assuming one's state of life or condition should completely affect how they feel is unfair and unjustified.
She has her own limits, her own feelings, and her own work management abilities, just like everyone else. It's a punch in the gut for you to dictate that she should work the exact same as someone else, just because that other person has a little more weight on their shoulders. Sure, it can be a decent motivator sometimes, but more often than not it aggravates and probably hurts the person you say it to. As I said, I get this thrown at me fairly often, and every time it happens, rather than motivate me to do more or whatever it's 'supposed' to do, it makes me question my worth in the eyes of the person who said it to me. Since it very literally compares my worth to the worth of someone else, and very obviously berates mine in favor of that other person's, it's a very painful feeling and often results in bouts of self-degradation.
I can't and won't speak for how your girl feels, but I did want to put my own experience in the mix. Perhaps she feels similarly to how I do in these situations, perhaps not, but I think you need the perspective of how the "well this person has it worse than you and they manage, what's your excuse?" line can affect people.
tl;dr say you're sorry and support your girl. Whether you agree with how she feels isn't what's important right now, she's obviously stressed and could use some comfort from the person who's supposed to be there for her, not jokes and ridicule made at her expense.
And also, remember the three golden rules of comedy:
Consider your audience and their environment.
Consider your timing.
Consider your wording.
Since you appear to be like me, the guy who uses comedy as a conversation-starter/tension-diffuser, adhering to those rules is an essential part of not pissing people the fuck off or inadvertently attacking their feelings.