Do birds know their last year's offspring?

Birds – ducks or sparrows, say – must obviously recognize their latest offspring. But if mom and dad sparrow bump into last year’s sparrowlings, now long fully grown and probably themselves procreating, will they know them?

Are there any species of bird that cultivate a kind of family relationship after their children flee the nest?

Yes, quite a few birds show kin recognition.

Some birds are even cooperative breeders, and last year’s brood may stick around to help mom and pop raise this year’s siblings.

Another, well-known bird that seems to be a cooperative breeder is the American Crow. From this page, http://birds.cornell.edu/publications/livingbird/spring98/crowsSp98.htm