Even after the death, only shrAdha ceremony will help the soul to reach heaven. The soul will mostly reincarnate in same blood line so can we conclude the relationship between father and son is eternal. Can we say relationship with son is only going to help even after death and that is the way to attain immortality, so it is eternal ?
In a way yes. Their relation is eternal because son is said to be the own self of the father, his atma.
angad angat sambhavasi
atma vai putra-namasi
sañjiva saradah satam
“You have taken birth from my various limbs and have arisen from my very heart. You are my own self in the form of my son. May you live through a hundred autumns.” This verse appears in the Satapatha Brahmana (18.104.22.168), Kaushitaki brahmana and the Brhad-aranyaka Upanisad (6.4.8).
atma vai putra utpanna
tasmad asya bhaved vakta
The Supreme Lord said: The Vedas instruct us that one’s own self takes birth again as one’s son. Thus let Romaharsana’s son become the speaker of the Puranas, and let him be endowed with long life, strong senses and stamina.
Thus, there are some injunctions in the shastras which show that the self itself takes birth as a son. Hence the relation is of soul and not of flesh alone.
The Manu Smriti also says the same. That the father himself gets born again as his own son. So, in that way the father-son relationship is eternal.
Manu Smriti 9.8. The husband, after conception by his wife, becomes an embryo and is born again of her; for that is the wifehood of a wife (gaya), that he is born (gayate) again by her.
And, similarly the husband-wife relationship can also be considered as an eternal one.
Manu Smriti 4.184. Infants, aged, poor and sick men must be considered as rulers of the middle sphere, the eldest brother as equal to one’s father, one’s wife and one’s son as one’s own body,
No, the relation does not seem to be eternal.
Because, first the sanatana dharma accepts no relation of body and soul. Soul is eternal as per the Gita.
2.20 Never is this One born, and never does It die; nor is it that having come to exist, It will again cease to be. This One is birthless, eternal, undecaying, ancient; It is not killed when the body is killed.
By Moksha soul is liberated. Moksha is the fourth purusartha. If we accept the relation of father and son to be eternal, none could attain moksha, as moksha ends the cycle of birth and death.
The sloka that says that AtmA is born as ward means the JivAtma, whose part is the sthula sarira, because the sperm is a part of the physical body. AtmA means a person or body also according to sanskritdictionary.com.
Each of us appear with our own bundle of prArabdha. Sri Krishna says in Gita
6.41 Attaining the worlds of the righteous, and residing there for eternal years, the man fallen from Yoga is born in the house of the pious and the properous.
6.42 Or he is born in the family of wise yogis [Persons possessing knowledge of Brahman. (S. concedes that some rare householders also can have this knowledge, and he cites the instances of Vasistha, Agastya, Janaka and Asvapati of olden days, and Vacaspati and the author of Khanada of recent times.)] only. Such a birth as is of this kind is surely more difficult to get in the world.
So a yogavrastha is born in a yogi's or a rich man's house.This clearly means that the relation is not eternal.
SankarAcharya also says:
kA te kAnta kas te putrah, ie you have no eternal relation with your wife and son.
All these induce me to conclude that the relation is NOT eternal and the inner meanings of the lines quoted by others is different from their outer meanings.