This is an interesting question as many people have heard this or similar things. The answer is yes and no.
The Mahanarayana Upanishad (translation by Swami Vimalananda, the commentary is an untraced author,but the commentary closely resembles Sayana’s commentary) says in section 79, verse 8 commentary. From the extensive commentary on this verse, it says:
The Taittiriyasamhita (VI.3.10) contains the famous statement that a man is born with a congential burden of a threefold debt—towards, the gods, the departed ancestors and the seers. This idea is elaborated later in traditional codes at length...Everyone owes a third debt [the first two to the gods and seers] to the biological ancestors culminating in their own parents. The sacrifices which a good father makes in order to raise and rear worthy children are impressively portrayed in legend, fiction, poetry and history. A worthy sone is exonerated from his debt of having received the protection and care of his sire only when he confers the same on his offspring. It is, therefore, stated here that procreation is the foundation of society and that only a person who rears children rightly in the way approved by the scriptures and who allows no break in the continuity of his race is absolved from this kind of debt.
And in section 59, verse 1 (which is an oblation to be made while making a sacrifice to Agni) the commentary says:
...Each person is not only responsible for his own sins of omission and commission, but also for the sins of the other persons who belong to him, or with whom he has relationship through blood, or over those whose actions he can legitimately control.
These verse are written to try and kindle a person’s moral consciousness and spur them onto a path of purity and perfection, the path of dharma.
At the same time, one should remember the words of Krishna in the Gita; specifically verses 9.25 and 18.65-66. (Swami Nikhilananda translator):
9.25 Those who worship the gods go to the gods, those who worship the manes go to the manes, those who worship the spirits go to the spirits, and those who worship Me come to Me.
18.65 Fix your heart on Me, give your love to Me, worship Me, bow down before Me; so shall you come to Me. This is My pledge to you, for you are dear to Me.
18.66 Abandon all dharmas and come to Me alone for shelter. I will deliver you from all sins; do not grieve.
If a person is leading the life of a householder, they should recognize the debt to their parents and ancestors by following dharma. And although you may lead a way of life and perform sacrifices on their behalf, one should recognize that worship belongs to God alone. Think of the Lord alone, try and fulfill you dharmic debts – but do not become fixated on them. By fixing your heart on the Lord, the Lord will deliver you from any sins.
By the way, even sadhus recognize their debt to their parents and absolve anyone else’s debt to them by performing a shradda ceremony for their parents and their own person at the time of initiation into sannyas. Oftentimes before consenting to initiation, a teacher will ask - 'What about your mother?' meaning who will care for her if you receive sannyas?
What are the specific karmas associated for doing/not doing or are extended to you or your parents/ancestors? The means and extent of any karma is known only by the Lord as the Lord alone is the dispenser of karma. (Brahma Sutras).
