Yes, you absolutely can. In fact, the more you expose yourself to an irritant, the more likely you are to develop a more severe allergy. I got hives from eating some weird pickle brand as a kid, and kept eating them anyway. My microbiologist/medical dad was like, "NO STOP THAT---YOU COULD TRIGGER A FULL-ON ALLERGY"...My thought process at the time was that I only had like 3 hives on my wrist and the pickles were good, so..keep eating....Apparently, that is a terrible choice lol! I did see his logic and stopped after he scared me about the risk of ignoring even minor hives for about a week.
They also say that no one is truly NOT allergic to poison ivy, but that some people just have an off-the-charts exposure threshold, to where they will never react (even with repeat lifetime exposures) that having been said, ENOUGH exposures of poison ivy would eventually cause a reaction in anyone. So, for one person, that may mean contacting the oil a few times, whereas for another, they would need to sleep in a bed of it nightly for 10 years to develop a reaction. The point is, repeat exposure to an irritant often leads to a reaction and reactions often develop or change based on age (but also can be impacted by underlying immune issues). This is why some people (despite exposures ) never get it in their lifetime and claim to not be allergic. The problem is, you never know when that next exposure could tip your threshold and result in an extreme reaction. I am one of those people who could say "I'm not allergic" to poison ivy-- so is my dad, but with enough forced exposure, I would eventually have a reaction (that is why you should still avoid it, even if you haven't had a reaction to it on a past exposure).
Parrots produce dander, and some parrots produce a lot more than others. Powder down birds like tiels, toos and grays are the worst, but even budgies (non-powder down) still produce a dander that can trigger allergies.
Do you have a really solid/expensive, non-ionizing/non-sanitizing true hepa air filter running in your bird room?