It's not always easy, but letting this stuff bother you can't be the focus. There will always be competition, and there will always be those who are critical, don't agree, and no matter what is accomplished by you or your athletes, that will not change.
Instead, take the opposite approach, and EMBRACE the competition! Solidify your successes, and be the villain in the critic's world! Be the one they want to beat. More success, more vindication.
It's a compliment really, because if you're doing things right and well enough, someone is not going to like you. In fact, if someone's not disliking you, you're not doing things well enough.
Embrace the competition, and be the villain in the critic's world.