Yes, as long as it is something they can normally sell (they sell parakeets) they can take them in as surrenders. Not if they need medical attention though (normally). They will then adopt it out to someone who can take good care of it.
No, Petco is not an animal shelter and outsource for their small critters (reptiles, mice, birds, fish, etc). They do have adoption days where a local animal shelter will bring animals (normally cats & dogs) to the store to be adopted. Try posting an add for the bird on CraigsList or a local FaceBook buy/trade/sell site.
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