If I were you, I wouldn't bother justifying myself to this person or otherwise explaining why your tiredness is actually valid. I don't know why so many people must make a contest out of everything, or belittle what others go through because it is different than their own circumstances.
Some kids sleep well, some kids don't (and my 3 yo still wakes up at night). Some parents are tired, some aren't. Some people have kids, others don't... however, I assume people are capable of empathy which should give them the ability to somewhat put themselves in someone else's shoes!
Sure, this person doesn't know you well enough to know about the fact that you care for an elderly parent... but this just means the comment is that much more out of place! Even if you weren't caring for someone with ailing health, you would still be entitled to be tired and celebrate a rare night of good sleep!
In a similar vein, I'm a parent and I get upset when people get all "Oh, but you have kids so you must get *whatever perks*.". Yeah, I have successfully reproduced... I don't think it should entitle me to skip lines, arrive late at appointments, make messes at restaurants, etc.