I’m a double legacy (both my parents went to the same college). However, my first choice is this college’s #1 competitor. I hope admissions at my first choice doesn’t see that both my parents went to their competitor and assume that I’m going to end up there too. Especially since demonstrated interest proves that “will the student choose to go here” matters so much. I thought about not including my parents’ alma mater in the common app but I don’t know if that’s a good idea either. Am I overthinking?
Most schools supplemental questions asked the highest level of education of the parents, not where they went to school. If you happened to be a legacy at a specific school, then they asked to include the relationship. So, your #1 won’t know your parents went to their rival. Also, if you apply ED, you are showing your love to your #1.
It won’t matter.
Well it might if the colleges in question are Auburn & Alabama.
I have heard a lot of justifications of why someone didn’t get accepted, but this one is out in left field.
Unless is Ohio State and you want to attend Michigan.
Just kidding OP it wouldnt matter. In fact they may enjoy torturing the legacy parent who has to send checks their way twice a year.
I went to Michigan, and they might find it entertaining to submit the spawn of graduates of THE Ohio State University. 
They’d assume it might make the parents unhappy.