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Generally, mites that affect birds do not migrate to your furry pets or yourself. I understand the heebie-jeebie factor when we see parasites on our pets. Rather than trying everything to treat this on your own, and spending a lot of money in the process, you should either take the bird to an avian vet or take a peice of scotch tape and try to catch one of the mites on it, then take that for identification. If you know the type of parasite, it is much easier to tell you how to get rid of it. As for why you have bugs under your skin, that sounds like chiggers, which is possible. But that's a whole nother story. Knowing who the enemy is, is the way to win this battle.
The way I found out about the mites is my bird was sitting on my shoulder and the mite fell on my chest. It is tiny and black with thin legs.
I know they are not bedbugs, I believe they are brown
Without seeing it, I could not make a diagnosis. If you still have it, and got a good look at it with a magnifying glass, you could look for some websites that can help you identify this mite. First off, you should identify if it even is a mite. How many legs does it have? Is the body segmented? Hav you seen more on the bird?