I realize that parents are exactly that: parents. It's only natural that they feel some sort of protective obligation. However, as kenpochad stated, this can become disruptive, and the last thing you need as an instructor, are more disruptions.
I've encountered this in the past, and the way I've dealt with this depended on how severe the disruption would be. If a parent were simply trying to give some minor coaching tips along the way, I'd wait until after the class finished, and speak with the parent afterwards. While keeping everything polite, and even casual, I would simply remind them that it was my class, and that I would ask them to simply wait until after the class was over, to talk to their children. I would also tell them that it's important to keep up the flow of the class, and that distractions, no matter how well intentioned they were, were to be avoided. Later on, when my assistant had passed her assistant instructor's exam, I'd have her handle a few of these cases (just for the experience).
Almost every time, this worked just fine, and the parents would refrain from interfering with the flow of the class. The few times it didn't work, the parents would either stop coming to the class (fine by me), or withdraw their child from the program. Going to another instructor's class wouldn't help them, since all of us had the same policy; no interference from the peanut gallery.
In more severe cases, where a parent would actually start screaming at the kid from the spectator's area, I would ask my assistant instructor to come out, and continue to teach my class for a short time, while I immediately went over to the spectators' area, and had a frank discussion with the parent. I'd emphasize the same statements above (e.g. not disrupting the flow of the class) in as polite of a manner as possible.
Sometimes this worked fine, and the parent would stop badgering the child during my class. Sometimes, it didn't work, and the parent would keep shouting at the child, despite my previous request. In those cases, I'd ask the parent to leave the area, since not only were they distracting the child, but also disrupting the entire class, and creating a spectacle amongst the spectators.
In all of my time as an instructor, I've only had one case of outright hostility from a parent. This parent had actually criticized my instruction, calling it incompetent and stupid, in front of the entire class, as well and the other folks in the spectators' area. At that point, I simply ceased having any more conversation with her, and continued to teach my class, since she wasn't in the right frame of mind to discuss these matters. Since the chief instructor was out of the country at the time, I had to wait until after he got back, to speak with him. The rather offensive lady had written a couple of nasty letters to him, claiming that he was doing the entire school a disservice by keeping me on the staff, and even making some rather outlandish claims.
When he got back in town, the chief instructor simply asked me for my side of the story, and also asked my assistant instructor what happened. After he heard the other side of the story, he simply came to the conclusion that it wasn't worth having this lady's child in the school anymore, and tore up the kid's contract. While we all missed having that child in the school (pretty darn good karate-ka, dedicated, and strong), we all figured that addition by subtraction was the best policy here.
Bottom line: When you teach, there's going to be a chance that you'll run into someone like the above. Take everything they say with several grains of salt, remain calm and collected, and always remember, it's nothing personal. If something comes up because of this, simply tell the truth about everything that happened, and let things sort themselves out.
I'll confess this much: for a few microseconds, I was tempted to speak my mind to this lady. After all, instructors are only human as well.
