Several of the clubs are 21 plus. The comedy place and the Adventurer's club allow children, but it is widely frowned upon by other guests.
If you feel okay with your teen hearing raunchy adult comedy though, I'm sure you'd be allowed to take them in. When I was there in Feb, after 10 the AC became rated R at times.
There are also a lot of very drunk people there later into the evening.
As for the music clubs, I'm not sure but I think a handful of them let kids in accompanied by parents. I'd be wary of this too. When I was there in Feb, there were a lot of guys trying to pick up women in the clubs and they are usually pretty drunk. They are not going to necessarily realize that a teen is a teen.
My friend and I were physically assaulted on PI when we were 21 by two 30 something men who were not pleased that we would not go back to their rooms several years ago. I've gone back to PI since but only to the AC and only with my boyfriend.
So if you go, be aware of what is going on around you and use your best judgement. IMO, older teens will be fine at the comedy and AC, but just be wary of letting them out of your sight in the music clubs.
Have a good trip!