I'm sorry you might have to re-home your Parrotlet, that's a tough situation..There is a specific forum here for "Re-homing Birds" that you could place an ad in. Make sure you explain the situation, where you're located, etc. The moderators can make sure that any member who might be interested in adopting him from you can PM them and they will forward the PM to you if you don't have enough posts to accept PM's yet.
As far as "boarding" him while you're on vacation, you just need to search for Certified Avian Vets or Avian Specialist Vets in the Tucson area, they are the best places to board a bird...Also, there may be some Bird Shops that also board birds in the Tucson area, so be sure to search for them as well.