How would rejection from both parents affect a child?
How would they act? what would their personality be like? And the most important question of all: What would help them get rid of or at least lessen these effects?
The psychological theory most commonly used to describe these sorts of situations is Parental Acceptance-Rejection Syndrome. Most of this research was pioneered by Ronald P. Rohner at the University of Connecticut.
The best book on the subject is probably New perspectives on family. The warmth dimension: Foundations of parental acceptance-rejection theory by R. Rohner (1986).
Here are some highlights from his research
For example evidence shows that the effects of parental rejection and emotional abuse are not limited simply to individual personality and behavior disorders, but they reach into such abstract domains as the religious beliefs of a people, their art, music, and other expressive behaviors. There now seems to be little doubt that the results of parental rejection as well as the mechanisms which produce or distort parental warmth work uniformly throughout our species, regardless of differences in race, nationality, time or other limiting conditions.
The main difficulty that a child who has experienced total parental rejection will have is an inability to give or receive love. Specifically, because the child has not learned strategies for how to emotionally and socially process unconditional positive regard, they will be at a significant disadvantage should it appear in their life, much like how a child who never learned French will have difficulty learning it as an adult. Children in this scenario tend to hide their emotions. They will, in many cases, cease to try to get affection from people who are important to them. Emotional attachments will inevitably be constrained by emotional constriction, or defensiveness.
Beyond a certain point, the dependency response may be extinguished or transformed. The seriously rejected child has not learned how to give love, because he has never known a loving parent after whom he can model his own behavior, and for reasons described below, even though he craves affection he has difficulty accepting it. In order to protect himself from more emotional hurt, the rejected child tends to wrap his emotions in cotton, to encyst his emotions. Ultimately he may stop trying to get affection from the people who are important to him. That is, dependency responses disappear. Thus the rejected child becomes emotionally insulated, unable to freely and openly form warm, lasting, intimate relations with others. His attachments tend to be troubled by emotional constriction or defensiveness, and in extreme cases the rejected child may become apathetic or emotionally bland or flat. In addition, as a result of the grave psychological damage brought about by rejection, the rejected child is inclined to have less tolerance for stress and he is therefore likely to be less emotionally stable than those who were accepted as children.
Children who are allowed to express anger without recrimination generally prefer that mode to becoming withdrawn. Totally-rejected children who find violence as an outlet will usually exploit it.
The rejected child is especially likely to become hostile, aggressive, or passive-aggressive if rejection takes the form of parental hostility. Under these conditions he is provided with an aggressive model to emulate, and thus his own aggressive responses may intensify.
Children who experience total rejection generally see the supernatural, if they believe in it, as a hostile, terrifying force. This includes perceptions of religion.
In general, rejection is experienced, and processed, similarly across all races, nationalities, times, or limiting conditions.
In terms of how to overcome such trauma: Abusive parents tend to live in isolation themselves, cut off from an effective network of friends, family, or other community support. It is sadly quite common for abusive parents to simply be overwhelmed, either with too many children, too little emotional, social, or financial support, or a terrible combination of many of these factors.
Social workers and clinical psychologists are best equipped to handle patients who suffered from such abuses. Providing social services, both physical, tangible benefits (food, shelter, education, healthcare) and intangibles (companionship, romantic love, open and honest communication) do a great deal to help someone who has been abused. The best strategy is unconditional positive regard. That is, to accept and love without judging or feeling a need to dissect, rehash, or understand. This is a sort of ego-less interaction that can be very draining on the provider.
In order for the patterns of behavior that develop from systematic abuse to be ameliorated, the subject must be treated kindly consistently for a long period of time so that the defensive habits can be unlearned.