Having the option to ditch offspring when conditions are too harsh to sustain them is one of the survival advantages of being a marsupial, rather than a placental, mammal. A starving mother kangaroo can leave a joey behind, save herself, and live to breed again when conditions improve.
Even today, some parents may willingly give up custody of their children if they cannot afford to raise them. Many countries have "safe haven laws" where a parent can leave a child (up to a certain age) at a hospital, clinic, police station, or firehouse and not be charged with child endangerment.
In Nebraska the original safe haven law allowed any child under 18 to be left at a hospital. Resulting in 35 non-infant children being left at hospitals.
In some cases the only way a parent can get a dangerously mentally disturbed child the help they need is to surrender their parental rights to the state.
Numerous studies with monkeys has put forth the theory that not only is there a problem with outright neglect, but even emotional coldness or social isolation can create socially disordered children later on. The same studies found that a surrogate would produce strange but okay children. So, basically, it's better to be Raised by Wolves than have Parental Abandonment.
Some animals, such as elephants and orca whales, learn how to raise young by watching family members do it. If a captive individual doesn't get this opportunity, it greatly increases the chance of the animal not caring for her offspring.
Brood parasitism, as practiced by several kinds of birds and fishes, occurs when the parasitic species lays its eggs in the nests of another species. The parasite's young are cared for by the nest's unsuspecting owners, and never meet their biological parents.
Back in the 1940's, René Spitz discovered that the infant mortality rate in the foundling hospital he was in charge of was far higher than the national average infant death rate. After trying several physical ways of fixing the problem, he discovered that the cause was simple lack of physical comfort, caused by the absence of a parent to care for the infant. When he had the nurses act as parental substitutes by taking time to simply cuddle the infants, the death rate dropped dramatically. See a silent film on the subject here