[Blueface]

Not an uncommon HOA issue.
Not sure the one I would pick a fight over. If in bylaws, you will lose.
If not in bylaws then let the games begin with a response from an attorney to them pointing that out.
However, guess what will then happen to bylaws? They will be amended to fix that void.
If you really want to display your flags and avoid a piss match, likely costly for you twice over as you will pay your attorney as well as association's via your maintenance fees, I say get a pole for them that works for everyone.
Unfortunately, HO associations are both good and bad for one's health.