Abuse of Psychopathic Kids
Psychopathy is mostly genetic. Therefore, it stands to reason that children of psychopaths would be more likely to be psychopaths themselves. They are also more likely to have suffered abuse or neglect. Although a few psychopaths report feeling empathy for their own children and no one else*, many more feel no more concern about the welfare of their children than they'd feel about an ant.
Liane Leedom's book Just Like His Father, which I reviewed here, is written primarily for parents concerned about the inheritance of psychopathy and antisocial personality - parents with unpleasant ex-partners, or adoptive parents whose children have biological parents with antisocial tendencies. Unfortunately, her confusion over the relationship between antisocial personality and psychopathy makes her book less useful than it could have been, but it's interesting that she recognized the potential link.
Parental Stress/Vicious Cycles
Psychopathic kids are very hard to parent, and my sympathy goes out to anyone caring for a child with psychopathic traits. Unfortunately, the behavioral characteristics of psychopathy can cause these children to bring out the worst in their parents.
Parental stress is one of the major contributing factors to child physical abuse. This study, focusing on parents of oppositional kids, found that several were being abused by their parents, and the abusive parents reported being under a lot more stress than non-abusive parents. Part of this stress was dealing with an oppositional child, obviously, but the abusive parents were also more likely to be under environmental stress, as measured by number negative life events in the past year. This shows that everything in a parent's life matters, and if you're getting overwhelmed by it all, your parenting suffers as a result. (This is also one reason why we should approach abusive parents with less blame and more help.)
Psychopathic kids also tend to elicit specific behavior patterns from parents which can have a harmful effect as well - specifically, by amplifying psychopathic tendencies. Firstly, punishment insensitivity, which is a characteristic of psychopaths, tends to elicit harsh punishment from parents (if the kid doesn't care about time-out, maybe a spanking will get his attention?) and this harsh punishment, in turn, increases punishment insensitivity. The result is a vicious cycle that could lead to parents crossing the line into abuse in the hopes of finding a punishment that will actually work on the child.
Parental attachment and child callousness is another area that could have a vicious cycle. Parents will get the sense that their psychopathic kids don't really care much about them (which is true, in many cases). The natural reaction from parents is to withdraw emotionally, both to avoid being hurt and because they feel it doesn't matter to the child. However, many psychopathic kids can form attachments to others - it's just a lot harder to do so. If parents give up on bonding, the kid will never learn how to bond with them, and will instead shut off whatever ability to bond he or she did have. This sets off a vicious cycle where the child's coldness gets parents to be cold back, and their coldness makes the child colder.
The last way that psychopathy may put kids at risk of abuse from a non-psychopathic parent is because of the stigmatizing effects of identified psychopathy. If the parent knows the child is a psychopath, they may use this to justify actions they would not have even considered with a non-psychopathic child. Online, I've heard many people suggest extreme measures for dealing with hypothetical psychopathic kids - on Yahoo Answers, the question 'psychopath as a child, what would you do?' had 'accidentally push him off a cliff' rated as the best answer. I have no idea how many people would actually act on this if they were in that situation, but it is a worrisome possibility.
Psychopathic kids may also be at risk of being hurt by people outside the family, because of their delinquency, risk-taking and lack of fear. All of these result in a kid who is likely to put themselves in dangerous situations without realizing just how dangerous they are.
Firstly, psychopathic kids (like many delinquent kids) are often poorly supervised**. This means they're often without any parental protection because parents are unaware that they've gotten themselves in danger. Psychopathic kids have a poor ability to judge danger both due to their young age and their psychopathy (fear is adaptive, it draws your attention to signs of danger and makes sure you notice them) and when they're unsupervised, their parents' ability to judge danger can't help them out. A child alone also attracts abusers, and though they prefer kids who give off 'victim' body language (which psychopaths rarely do), they'll gladly go for any kid they think they can take advantage of. Some of these abusers may be adult psychopaths, others are just messed up. Psychopathic girls may be especially at risk, since many abusers prefer to target girls.
Delinquent activities also put the kid in direct contact with dangerous people. In particular, many delinquent kids may consume products (drugs or hookers) provided by organized crime, and organized crime members can be extremely dangerous if you threaten their business (or sometimes even if you get involved at all). A psychopathic kid may not realize this, or may be blinded by the thought of the rewards they could gain by screwing over or working for organized crime members, and quickly get themselves in deep trouble with connected people. Even adults often don't realize the risks - kids certainly won't.
Psychopathic kids tend to lie a lot. And, in particular, many make false allegations of abuse, as revenge or manipulation or just to get a reaction. One serious consequence, once a kid has gotten a reputation for false allegations, is that it may be very hard for that kid to be believed if they really are being abused.
This is why, no matter how many false allegations a kid makes, you can't just ignore their allegations of abuse. The 'crying wolf' story is taken as a parable for the boy who cried wolf, but it's just as much a parable for the townsfolk - just because someone sends out false alarms does not mean all their alarms are false. While it's important not to cause too much harm to the falsely accused, all abuse allegations need thorough investigation, even if they come from a child who has made false allegations in the past. To do otherwise risks abandoning a child to be abused.
Does It Matter?
There is a stereotype that psychopathic kids aren't hurt by abuse. Certainly, they seem less responsive to poor parenting than non-psychopathic delinquent kids, their fearlessness can protect them from developing trauma-related anxiety disorders, and their grandiosity may protect them from abuse-related shame. But this does not mean abuse has no effect on psychopathic kids.
In particular, there is good evidence linking abuse of psychopaths to the psychopath's likelihood of becoming violent. Some psychopaths are never violent - they con and use people, but they never actually lay a hand on them. Others may be severely violent, and a rare minority may even be serial killers. One of the predictors of whether a psychopath will be violent or not is how much violence they've been exposed to. (I'm not saying this is the only predictor, because there are multiple pathways to violence, but it is a major link.)
The thing is, though psychopaths may have reduced feelings of fear and possibly sadness, they are just as capable of getting angry as anyone else - if not more so. A psychopath who has been abused has a lot to be angry about. Furthermore, abuse teaches psychopaths a way of behaving, making them more likely to use abuse and violence as a conscious strategy.
* See this thread. It's possible they're lying, of course, but it seems plausible to me, considering how many species of animals care only for their young and no one else. Parental concern is one of the most basic forms of empathy.
** This often has more to do with the child's behavior than the parents'. It's hard to keep track of a kid who lies about or doesn't tell you where he or she is going, skips school, breaks curfew, and sneaks out. Well behaved kids are a lot easier to supervise properly because they'll tend to be where they're supposed to be.