In Hinduism, it is generally said that children too have a share of their parent's sins & good deeds.
So, according to Buddhism, does the karma of a parent (in a direct or indirect way) affect their children?
Karma of one person is not shared or inherited. Say a parent has a bad Karma to suffer. This might some times instrumented by means of a child getting sick. This does not mean the Karma has being pass to the child. Say the child leaves the household then perhaps this may manifested through another calamity even maybe another loved one getting sick.
Karma is Fabrications. The volition creates fabrication giving, pleasant, neutral, painful experiences in the future. This is Karma. This cannot be transferred or inherited or stolen. When this gives results there should be a medium for it to manifest. Generally though the people and the environment around us. The working of Karma is complex. There could be a case some one dear might die so we experience sadness. In such a case it is not the Karma of the person who died inherited our bad Karma. If there are X & Y who are dear, if there is a chance X may go and in case the opportunity does not present itself for X and Y is there then Y maybe the person to die. This is not X or Y's karma but our own which manifests itself to give pain. If say the opportunity of you get hurt though X & Y does not present itself as the environment and situation did not arise, it does not mean the Karma will also follow X & Y but it will follow you and will give results at a later time if it is potent and the opportunity present itself. In this case if is Z how is close and dear to us and the karma is that for a loved one to depart it might be Z who takes the fall.
Also it is not wise to be around people doing or having bad Karma, or situating where bad Karma is in effect. E.g. if you see a rioting or fighting crowd get your self out of there and if you have the slightest part Karma to get beaten up this might be the best opportunity for it to manifest itself. Also if you do not have such Karma getting being at the wrong place at the wrong time, may mean you can get hurt.
According to Buddhism, there are five orders or processes (Niyama Dharma) which operate in the physical and mental realms.
Utu Niyama - physical inorganic order, e.g. seasonal phenomena of winds and rains. The unerring order of seasons, characteristic seasonal changes and events, causes of winds and rains, nature of heat, etc., all belong to this group.
Bija Niyama - order of germs and seeds (physical organic order), e.g. rice produced from rice-seed, sugary taste from sugar-cane or honey, peculiar characteristics of certain fruits, etc. The scientific theory of cells and genes and the physical similarity of twins may be ascribed to this order.
Karma Niyama - order of act and result, e.g., desirable and undesirable acts produce corresponding good and bad results. As surely as water seeks its own level so does Karma, given opportunity, produce its inevitable result, not in the form of a reward or punishment but as an innate sequence. This sequence of deed and effect is as natural and necessary as the way of the sun and the moon.
Dhamma Niyama - order of the norm, e.g., the natural phenomena occurring at the advent of a Bodhisattva in his last birth. Gravitation and other similar laws of nature. The natural reason for being good and so forth, may be included in this group.
Citta Niyama - order or mind or psychic law, e.g., processes of consciousness, arising and perishing of consciousness, constituents of consciousness, power of mind, etc., including telepathy, telaesthesia, retro-cognition, premonition, clairvoyance, clairaudience, thought-reading and such other psychic phenomena which are inexplicable to modern science.
The Buddha says:
"I declare, O Bhikkhus, that volition is Karma. Having willed one acts by body, speech, and thought." (Anguttara Nikaya)
Every volitional action of individuals, save those of Buddhas and Arahants, is called Karma. The exception made in their case is because they are delivered from both good and evil; they have eradicated ignorance and craving, the roots of Karma.
According to the Niyama Dharmas, the 2nd which is "Bija Niyama" does indicate a determinant force which is inherited by the parents. But in my opinion, it is not a hypothetical idea that what our parents do right here right now or in the far away past, affects us today in the present! No it does not.
It is more about what we inherit - by that I mean our genetical inheritance. And that is predetermined by our own Karma : which is the 3rd Niyama Dharma - "Karma Niyama".
Our actions bear results. And they are partly what we inherit in a next existence.
So the answer goes as such All that we experience begins with thought. Our words and deeds spring from thought. If we speak or act with evil thoughts, unpleasant circumstances and experiences inevitably result. So have pure thoughts, and the results will be pure! Nothing to worry about our parents' deeds!