When a person takes to sanyaasa ashrama formally, as a part of the rituals, he is required to remove the sacred thread and discard.
This is to symbolically indicate that the person need not spend any more time doing religions activities, since all religious activities (like sandhyavandane, pooja etc, including pitru tarpana/shraadha) need the scared thread. Instead, the focus shifts to obtaining Brahma jnanam if not already obtained, and then the singular goal is the preservation and propagation of the scriptural knowledge. This person is now expected to have a minimalistic living (only kaashaya vastra, no hairdo, alms for food etc), hence there is no bank balance even if the sanyaasi wanted to do rituals.
Thus with the formal acceptance of sanyaasa, the person is free from the all karmic debts or rina, not only with parents, but with the society at large. That doesn't mean his love for his purvaashrama parents is gone. On the contrary, due to Brahma jnanam, he has the same love and compassion with the entire world and all beings (not just human beings)
Then what about the parents? Who will do their shraadha etc after they are no more?
Shaastras say, the amount of punyam obtained by the parents by giving birth to such a nobel, selfless person who has dedicated himself to help the society (loka sangraha) far exceeds the punyam generated due to the performance of shraadha.
One more question may arise: If a sanyaasi is not required to do rituals and pooja, then why do Swamijis do pooja? For e.g we see Sri Bharati Teertha Maha Swamiji doing Pooja live every night on Sankara TV?
The answer to that is Swamiji does not need any rituals and is not doing the Pooja for his sake. He is doing it for our sake. The masses follow whatever their leader does. So if Swamiji stops doing pooja, the followers will think even they don't need to do any pooja! Hence Swamiji does religious activities only for loka sangraha.
(I'm not able to specifically quote sources for all this information, as this is something that I have gathered while listening to many many classes on Vedanta from my Guru, Swami Paramarthananda)