I think this situation divers from person to person. Now granted, I am only 19 years old so I can't really speak of having kids yadayada, but here's my two cents.
Some kids are mature enough to decide on a tattoo when they are 16, whereas others are 20 and still have the mindset of a 12 year old.
I personally consider myself mature for my age, and I honestly always have, which is probably why I was patient enough to over think my own decision of getting a tattoo and neatly waiting until I was old enough.
I think the best thing we (and by we I mean both admirers and artists) can do is to be open and advising to whomever wants to get a tattoo.
There is no real way to say if you should allow tattooing of under aged kids because each kid has a different level of maturity and common sense.
I see a lot of people comment about kids choosing things (bongs, cartoon characters, weed plants, whatever) and then later regretting it.
However, I have also heard more than enough stories of proper adults choosing a design for a tattoo and later regretting their decision, thus covering the tattoo up with something else or removing it completely. (My partners mother being one of them)
I believe you can never truly make a proper judgement of someone's choice because we all have different reasons for our choices.
When/if my child shows an interest in tattoos I want to just be open about them with it rather than risking them going into some back alley to get some sloppy job done risking infections and the likes.