Is there a word for the length of time an animal stays with parents before going off on their own?
As you talk about bird, most of the terminology I am suggesting below comes from the ornithology literature.

Fledging period

Defined on wikipedia:


  Fledging is the stage in a volant animal's life between hatching or parturition and flight.


Nestling period

Defined in Kouba et al. (2015):


  In altricial birds, the nestling period is an important part of the breeding phase because the juveniles may spend quite a long time in the nest, with associated high energy costs for the parents.


Rearing period

See its usage in for examples Yuan et al. (1994) and Brickell et al. (2009)

Juvenile

While the term juvenile does not refer to a period of time, you can refer to this concept as the period while children are still juvenile. From dictionary.com


  a young bird in the stage when it has fledged, if altricial, or has replaced down of hatching, if precocial.