Honestly, fining children is only going to cause grief for the parents. However, if the fines were for community/school service, it would be a good idea. It would instill a sense of right vs wrong more prominently than what would be a vague and mysterious monetary fine that would have no bearing on a child who doesn't understand currency. I do think they're taking it too far in giving out tickets for "Misbehaving on the Bus" and "Profanity", but overall I do think the idea has merit.
Also, teachers ARE disciplinarians as well as instructors. Teachers can hand out detentions, etc. They have certain powers of discipline, it just depends on the teacher and school board, just how much EFFECTIVE power they have. A stern teacher that carries themselves properly will not be disobeyed.
Also, teachers DO spend more time with your children between the ages of 5 and 17 than parents do, and that's something I think most people tend to forget in this day and age. This actually creates a rift between parent and child that doesn't end up being bridged easily. Compound that with the fact that many parents work even while their children are at home, giving parents even LESS time with their children to help raise them. Like it or not teachers, your job IS raising children to be respectable members of society. You are CO-RAISING these children. Many parents are shirking their jobs, and don't do anything to raise their children and that IS a problem. But people, especially teachers, often say that it isn't their job to raise the children. Yes, it IS. Most classes start between 8 and 9 am, and don't end until 3 or 4 pm, depending on the school. So let's break down an average child's day.
An hour for lunch, and 20-30 minutes for recess, that means a minimum of 4 and a half hours a day and a maximum of 6 hours 40 minutes a day with teachers. Most schools being somewhere in the middle of these figures. So we'll say 5 and a half hours a day are spent in the teacher's care. That leaves 18 and a half hours left in the day. Children need more sleep than the average adult, so we'll give 10 hours a day to sleep. They like to play with their friends, we'll give 3 hours to that. There's homework, we'll give approximately 2 hours for that (averaging projects plus normal daily stuff), and I'd say that's underestimating. Lunch, and Recess, which is about an hour and a half. 18.5 hours left in a day, 1.5 for lunch and recess leaves 17. 10 hours for sleep leaves 7 hours, 2 hours for homework, 3 hours for playing with friends, leaves 2 hours A DAY. And if we include morning routine, supper, the possibility of the parents working during times when the child isn't in school, that reduces the time significantly. Even if the child spends only an hour with friends, that still leaves 5 hours or LESS with the parents per day. Yes, parents can help with homework and etc, but the point still stands. Teachers spend a LOT more time with children than their parents.
TL;DR, Idea has merit, but should be some sort of service. Teachers spend more time with children than parents do once school begins. Parents and teachers co-raise today's children. Teachers, it IS your job to raise children and be a disciplinarian. Parents do shirk their duties, but so do teachers, leaving children to raise themselves.