Best way to find out is to have an allergy panel done to see if bird dander is among those substances you are sensitive to. Based on the info you give, I would hold off getting a bird until after talking with your allergy specialist.
Evey body is different, it would depend on how sensitive you are, if you have asthma I would say it's a definite no, to cockatiel, greys, and toos. Most other birds are ok, again depending on your level of sensitivity.
I think with each case is different.Everyones body reacts differently.I used to be asthmatic along with allergies.My doctor told me I would never outgrow the asthma.Well I wasn't born with asthma either.It's been over 5 years since I had any symptoms of asthma and no medication.I would say stay in contact with your doctor.And I've always had Cockatiels,Cockatoos and the more dandery FIDS>