Marie has some good advice and answering her follow-up questions would be helpful.
The number one thing to know about bites is
avoid them. Birds give off multiple signals via body language before they bite so try to familiarize yourself with the body language and respect signals that say, "I'm going to bite." It's really, really easy to accidentally reinforce bites and once bites have been reinforced it becomes a learned behavior and it's hard to reverse at that point.
Yes, parrotlets can be aggressive and prone to biting but with proper handling you can work through it. It will not stop on its own and if she continues to be handled improperly it will get worse (learned aggression).
pamala1983 wrote:We have been telling her no bite and touching her beak when we say it. However, it seems to just make her bite even more and harder when she connects-she seems to almost go out of her way to try and bite now. What are we doing wrong?
Although I have seen a few people recommend this approach for biting it's totally wrong!
1) Just following a bite the last thing you want to do is present your hand for another bite
2) Touching the beak escalates the situation as it makes the bird
more aggressive (if you ever see two birds fight they spar with their beaks so touching the beak after a bite for all intents and purposes is "fighting" with the bird)
The correct response to a bite is to first, have no reaction to the bite whatsoever and second, immediately cease interaction and ignore the bird for a brief period of time by turning your back. Don't leave the room or place the bird on/in its cage or on a perch in response to a bite because that could accidentally be reinforcing.
pamala1983 wrote:We did notice that she is more aggressive when you are reaching near her cage like to pick her up or move her off the top of the cage.
Cage aggression is pretty normal but if you use positive reinforcement to teach "step-up" usually the bird will step-up even from its cage without biting.
Basics of parrot taming and training (including step-up)pamala1983 wrote:We have kids and sometimes they want to hold her but they are nervous around her and they jerk thier hands around her scared she is going to bite so I was wondering if we should just not allow the kids to hold her until we get the biting dealt with?
Yes, definitely don't allow your kids to hold her, at least for the time being. Once she is tamer and you have worked through the nipping it should probably be okay but jerky movements will startle even a tame bird and can always trigger a bite.
Hopefully that was of some help. It will take time and effort on your part but you can address this problem. If you have any more questions please ask.