If the coach won't address the problem the HU should.
Why should I as an umpire be responsible for a spectator and not the HC? Is that parent not affiliated with the team? Shouldn't the HC be responsible for his players and parents? Or better yet, why don't the "adults" in the stands speak up and self-police or if they really believe something is wrong, they go and get the TD or UIC?
What if my opinion differs from yours? What if I don't believe things being uttered in the stands are impacting a game? Or, what if my decision is to eject all of the spectators in the stands to insure no more interruptions? Would disagree with my interpretations of control of a game?
I have yet had to eject a parent. I suppose there is some scenario that could occur that might lead me to that, but I can't think of one at the moment mostly because as MTR already pointed out, I typically ignore what I deem as unhelpful remarks from any spectator. Furthermore, if I ever do, I would be wrong in doing so. I have lots of other tools that can fix the issue.
I watched that video. I believe my fellow official made a mistake by even turning around. I heard absolutely nothing that would have caused me personally to turn around or even address what was said.
If you have multiple games where you have witnessed a PU ejecting a spectator, I would charge that it's likely both a cultural and training issue from wherever you see this occurring regularly. If local rules state a PU has that authority, then so be it. You won't get me to agree with it though.