Multiple birthays problem

The classical "birthday problem" that we saw last week tells us that, in a room with at least 23 people, the odds are good that two of them will share a birthday.

Now what about three birthdays? Assume for this problem that everyone has exactly two parents, and they know both of their parents' birthdays. (And again, assume no one is born on Leap Day.) Wouldn't it be a crazy coincidence if you met someone and their birthday, plus the two birthays of their parents, were exactly the same as your birthday and those of your parents?

To be very clear, by "the same" I mean that the set of three birthdays is the same, not necessarily in the same order. So it might be that your birthday is the same as your friend's father's, and your friend's birthday the same as your mother's, and your father's the same as your friend's mother's - that would still count.

Given that the Naval Academy enrolls 4,576 Midshipmen, what is the probability that at least two Mids share this deep familial birthday connection?

(Hint: you will probably need a computer to do the calculation. I recommend Python because it handles big integers really easily. You can make some simplifications to ease the computation; if you do, be sure to explain what simplifications you made and why they're OK. Try to give a LOWER BOUND, like "the probabity that two such Mids exist is at least such and such".)