I would like to know if the Karma of parents has any effect on their children?
Or are they independent and have to realize their own Karma (prārabdha/sañcita)?
Yes, the Karma of parents does affect their children.. Here is what Brihaspati's brother Uchathya tells Vishnu's incarnation Mandhata (whom I discuss here) in this chapter of the Shanti Parva of the Mahabharata:
The tears shed by weeping men afflicted with falsehood slay the children and animals of those that have uttered those falsehoods. Like a cow a sinful act perpetrated does not produce immediate fruits. If the fruit is not seen in the perpetrator himself, it is seen in his son or in his son's son, or daughter's son.
But to that I would put one qualification: the family that you are born into is ultimately determined by your actions in past births, as I discuss here. So even if you are punished for your parents' sin, the punishment you experience may also be due to the actions you did in past births that made you be born into that family in the first place.
Children are born a couple based on the combined net effect of their karma. It is said in jyotisha sastra that the happening (often bad if any) child's first four years are result of mother's own karma, the next four years that of father's, the next four years of that individual.
Children can give and take life of their parents i.e. children born on amavasya when Moon is merged in sun, indicates, loss of maternal love & affection, hence considered to be a bad omen by some. Similarly children in purnima when sun & moon are bright reflects parents affection in plenty.
Moon represents mother, Sun represents father. If either of them afflicted then that karaka suffers, say like a child born during eclipse, depending it lunar or solar eclipse, one would suffer for Rahu would be in close degrees with Moon/Sun. Rahu = shadow
Every individual brings his own fate & destiny. It is a beauty of nature's law that using jyotisha sastra, with one horrorscope, presuming the jyotish is a learned one, one can predict three generations before & after ... i.e. one can predict on father/mother, grandfather/grandmother, children & grand children, prosperity of brothers/sisters.
Devaki-Vasudev prayed and got children, seven got killed by Kamsa, eight child survived (Lord Krishna) who gave moksha to his parents. However, the first seven children brought grief & sorrow to their parents.
Every soul of an animate or inanimate, has its own individual karma, plus its family karma (Bhagirath who brought goddess mother ganga to mother earth), community karma & state, country karma
Person born in America enjoys better privileges than an African born = country karma. Similarly extend it for other segments too.
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The short answer is "yes" and "no." A more descriptive "short" answer is that the karma of the parents in their past life affects the karma of the child, but the karma of the parents in the current life does not.
We are linked by karma to everyone else in the world, because the actions and abilities of others do, quite obviously and factually, affect us. When the jīva takes a body as the child of a certain set of parents, it is not randomly. Because the parents' karma determines their place in life, and that situation in life will affect the upbringing of the child and the child's current and future actions, behavior, attitude, conduct, and life situation, the child's karma is affected by the parents' karma in their past life--their "karmic debt" is affected by their place in life and their sufferings as a result of their parents' place in life. This is simply a logical deduction. If, of course, you have any objections to this, feel free to let me know, because I know this may be imperfect in several ways.
Wendy Doniger (controversial, I know) says in her book On Hinduism on page 104 (this book wasn't the one that was banned):
...the transmigrating soul is said to mix its own inherited karma with the karma of each of the new parents to form the cumulative karma of the new child.
To support this statement, she gives a reference to some other work or maybe even Hindu scripture in the endnotes--however, I am unable to access the reference because I could only obtain a preview from Google Books. I encourage anyone with the full version to leave a comment informing me of the reference.
However, what the parents do in the current life does not affect the karma of the child in the current life, except by indirect means (if the parents instill bad values in the child by setting a bad example, the child may commit wrongdoing later in life). But a theft, for example, by one of the parents will not affect the child. There is no reason to think it would. Philosophically, it creates some extremely obvious problems. As far as I know (and there's a really good chance I might be wrong here), the scriptures doesn't explicitly deny this idea, but see Bhagavadgītā 5.13:
sarvakarmāṇi manasā sannyasyāste sukhaṃ vaśī | navadvāre pure dehī naiva kurvanna kārayan
This may be translated very, very literally as:
Having [completely] forsaken all actions mentally, that which is embodied [i.e., the soul] happily remains one who has control (i.e., the controller) in the city of nine gates (i.e., the body), not doing, not causing [anyone] to do [anything].
Now if renunciation means that the soul becomes the controller, and no action will take place, then the parents' current actions cannot affect the child's current karma. Otherwise, religious renunciation could take place, but the soul would not become the controller and it could not be said to be in true "inaction" (as described in this verse) because it still would be accumulating karma. Thus, the parents' karma in the current life does not affect the child's karma unless the child chooses to follow in the steps of the parents and commit some crime.