Should a parent report their own children to the police if they are aware that
I don't think there is a hard-and-fast rule to give here. Do you call the cops when you see your kid litter? Of course not! Just make them pick it up and give them a good lecture about why that is unacceptable behavior. But if you see them commit murder? Well, yes, then it seems appropriate. If I caught one of my children shoplifting, I would try to come up with a way to make them repay the store--but I don't think I would be supportive if people at the store gave me an indication that they aggressively prosecute every case of shoplifting.
I think our responsibilities change in different relationships. I would also try to "correct" minor misdemeanors (like littering) when done by friends or more distant family members. The worse the crime, the more it seems to me to call for a legal report. But I think we are, in a way, much more responsible for the behavior of our minor children than we are after they have reached the age of majority, and we are much less responsible for distant relatives, acquaintances, and the like. So my own culpability in failing to report some law-breaking is relative to the degree of my responsibility for the behavior of that other person.
Do I call in every case I see of someone speeding past me on the freeway? No. That's not my responsibility. But if I see evidence that they are seriously impaired in some way (weaving dangerously, etc.), well, yes, I would call that in.
I think the only good advice I have to give here, beyond such rules of thumb, is that you exercise the best judgment of your own level of responsibility (to the criminal, to his or her victims, and to your fellow citizens) and of what you can do that is most likely to provide the best available resolution to the situation.