I support the ideas most have posted here including scubatoads. I am new to diving, my fiance was certified eyars ago and just did the refresher, and my 10 year old daughter and 12 year old son just completed their OW certifications in the Bahamas. The kids wanted to get into scuba because they love the water and wanted to be able to dive as a family (the both had their first swimming lessons on a regular basis by the time the were 6 months old.)
In the pool, I was in the water, and the intructor new that I was available if and only if he wanted me... which he did at a couple of different times. The instructor and I agreed before hand that I was not their to help the kids get through their certs, I was their to help him if he needed it. As for someone dropping their kid off for scuba lessons... I can't imaging it and don't think the LDS should even allow it.
As for their OW dives, I had the same agreement with the instructors and for the most part I I was just doing fun dives while the kids worked with the instructors. After they were finished I was told that they were as competant or more than many of the adult students that go through the program.
Now, I would NEVER ask another adult to buddy with one of my kids, and I will only dive with them in environments where I feel that I would be comfortable as a solo diver with the additional responsibility of making sure my own child is safe. As long as they enjoy the sport and the time together, I think it is great, but no parent should put that pressure on another diver.