Do you not have a floor fan or ventilation to pull the exhaled vapor out of the parrot's vicinity? Like face a floor fan toward the bathroom and turn on the exhaust fan? Pulling everything away from your parrot. Or crack a window (if you live away from people). We've always smoked around our dogs, with no problem, some of them enjoy the contact high and those who didn't would leave the room, it was their choice. The new dog we got was very afraid of the lighter, someone must have tormented him with one. In less than 3 months he's stopped running out of the room and will lay on the couch while we pass the pipe. Before I started growing my own, I had to use a vaporizer, because everything made me cough until I thought a lung was coming up, but my stuff's so smooth I can enjoy a pipe again, no coughing.
Where there's a will there's a way, so just think about it, and you'll figure out how to vape and keep your parrot safe.