To put it briefly... zoonosis. Very rarely can diseases, parasites, viruses, etc. just jump between species. The only time they are successful is if there has been enough time for that specific organism to acclimate to a new host, or if the hosts are very, very similar in the targeted infected area.
One of the few diseases that you can receive from parrots is Chlamydia. Not the STI, of course, but a different strain also known as psittacosis, parrot fever, or ornithosis. It has symptoms very similar to a bad case of pneumonia, which is why it's often difficult to diagnose. There's a few more, but that's the main one. It's caused by the contraction of that bacteria, so essentially if you don't keep the cage clean or let your bird touch your mouth with their beak you could be at risk of contracting it if you're unaware they have it.
