So what do you do?
What I do when my parrotlets go into the biting state or what a lot of people call the Terrible Twos is continue to work with them daily. Remember that each bird has its very own personality and you have to be the judge as to what your little bird can handle because you don't want to put to much stress on them. Keeping that in mind I always force them (if needed) out of the cage even if I have to catch them using a paper towel (no gloves), I move them to another part of the house where they can not see the cage. Otherwise they will not stop trying to get back to it. With the birds wings clipped I sit down on the floor with it and put him on my finger, He jumps off, I put him back on. ( you can also try using a perch in place of your finger) I do this over and over again for sometimes a strong 10 minuets until he gives up and remains on my hand. I also use millet treats holding them in one hand and letting him have the opportunity to have some each time he is on my finger. If I have a male or female that just no matter what will not give in, I cuddle them in my hand placed up against my chest as I continue to stroke his back with my fingers and talk to them with a sweet voice for 5 or 10 minuets and sometimes even walk about the house doing this. Normally they don't bite hard - However some of them do - In that case I use a paper towel to hold the bird (cuddling him) up against my chest with his head allowed to stick out of the paper towel and allow him to just bite the paper towel until he gives up (no gloves). All the time offering him millet treats and talking to him in a sweet voice. Remember as I stated above you have to be the judge of your own birds personality and stress level. Doing this the way I have described above has worked for me 100% of the time. I have never had a bird that did not calm down and get over this biting stage by doing this. I sometimes have success in as little as one day, and on some birds I have to repeat this process for 2 or 3 days. Parrotlets are have very strong personalities and can be very strong headed, You must teach them that no matter how you hold them you are not going to hurt them and no matter how much he bites the paper towel (or your finger) you are not going to give up.
Not all birds need this kind of training, Many times with parrotlets they simply outgrow this biting state and things go back to normal in a couple of weeks. But you do take a chance of the bird learning that all he has to do is run or nip you on the finger and he can get his way. That is something you don't want him to learn.
If your little guy was handfed as a baby this will work, it may take longer for each bird depending on the personality and how much it was socialized as a baby.
Do NOT - Just leave the bird in the cage and hope he will just some day get over it and run over and jump on your finger. It is very rare that will happen. If you stick your finger or hand in the cage and he runs away and you give up and let him continue to run away. You will loose your birds tameness completely.
Be very gentle when holding the bird, Do not squeeze him, talk sweet to him, offer him treats, cuddle him using a paper towel if needed (no gloves) and your little bird will tame back down in no time at all in most cases.