The Op post question was about installing spyware by parents not about bullying.
Sorry, got caught up in the side-bar.
Not really sure who is, or who isn't, a parent on here, but those who are probably have the same fear as I concerning their children and the internet. The internet is full of filth. Even Google images can be dangerous (Google "Cortana" (the halo character) and the pron starts in the fifth row!)
So, I don't think the question is really one of ethics, I think it's one of means.
A parent (IMHO) should do whatever they possibly can to 'spot-check' their children's activity on the internet. I personally have my Cisco router set up to block some things, allow others, and log the crap out of everything else. Most parents wouldn't know how to accomplish this, so if they can use a program like the above, then yes, they should.
It's abhorring to me that children under the age of 16-18 are assumed by any parent to have any kind of privacy. My children will not enjoy that luxury, because I brought you into this world, so I'll do everything I can to help you make the right decision until the law says I can't.
Now, I think the followup (and more important) question is, what should parents do with the information they get from spyware or the like?
Do you forcibly prevent further activity? Do you punish for the first (series of) offense(s)? Shitty parents worldwide take this approach, whereas any child abuse counselor will tell you that you need to talk to your kids about anything that happens, and explain the behavior, outcomes, effects, etc. and especially don't go in guns blazing. Kids (after adolescence) don't learn right from wrong simply from discipline anymore. You can't spank it away like you could at age 3. You have to appeal to the more logical side (in conjunction with an appropriate discipline for further offenses).
That's the real problem. We are treating children like adults from far too young of an age. We are assuming that 'they'll figure it out.' Umm, no. They won't. They'll draw inaccurate conclusions from your lack of direction and you've essentially accomplished nothing.
So I guess to sum this all up;
If you're genuinely interested in your child's development, then yea, install spyware, install keyloggers, check logs, and then teach (keyword!!!!) your child the dangers of any found actions.
However, if you're just another shitty parent who doesn't really care how your child fares after they're no longer your problem, then why bother? Don't install spyware just so you can become parent of the year by embarrassing your child, the one time you take an interest in their life.