Third person plural auxiliary for singular noun?

I was thinking about this today. In the situation when you have a singular choice but two options in a sentence, why would one use the plural auxiliary?

Example:

Have either your mother or father died? (Have = third ps pl, but singular noun, mother or father)

But then I've also heard:

Has your teacher or principal been to a conference?

I don't know which grammar rule would be correct.

I know that, from an American perspective, we make "native errors" all the time, which is why I'm confused.

Example: There's three dogs outside...when it should be there are three dogs outside.

Sorry...just the musings of an amateur linguist