Yes, it can! I don't think it would happen every time with every single rehomed adult parrot but it does happen and it's not restricted to new words for a parrot that was already talking and merely learned new words but it also happens to birds that never talked before.
I have an older female YC amazon (age unknown but believed to be now in her mid-fifties) that only spoke Spanish when she first came but learned words and phrases in English since then.
And a male blackcap caique that only said his name but now says 'Come on' when he wants to step up to people.
And I have a female quaker that never said a single word in her first home and now says her name and about 8 to 10 new words or phrases.
Parrots are not only very intelligent, they are also hardwired to communicate verbally with their mates, babies and flockmates so all it takes is consistency and repetition for them to learn to speak human.