Applying for Masters when Family Member is Associated with Department I intend to apply for a masters program in the coming months. A parent is a professor in the same field at a nearby institution and is also an adjunct in the department I am applying to.

Do I have any ethical obligation to "disclose" (or, for that matter, to avoid) this fact in my application? If I do not address it, could it be seen as trying to hide a conflict of interest? If I do address it, will it be seen as inappropriate name-dropping? If I must mention it, what is the proper way to do this?

I have an uncommon last name which will probably be recognized, and some members of the faculty already know that I am the child of their colleague.
I think you should say nothing.  If your name is uncommon and your parent also teaches in the same department, it seems certain the admissions committee will twig to the connection.  If they do that on their own, human nature being what is, you'll probably inherit a slight glow.  But if you make any mention, I think that will sound like name-dropping.