The Hindus had this idea of following in your parents footsteps. [I mean that they thought this is desirable.] To me this makes a lot of sense as long as what one's parents are about and what they are doing is not obviously wrong. I think one's parents and ones immediate family and one;s immediate friends have a lot to do with what one is. They not not just reflect on his inner essence but are apart of it.

See חידושי הגרנ''ט from Reb Naphtali (Troup) [One of the great Roshei Yeshiva at the time of Reb Chaim Soloveitchik.] He considers כיבוד אב ואם (honor of one's father and mother) as a regular מצוות עשה(positive command).  [I forget the context and I do not have the book to be able to look it up. His book incidentally is highly related to the חידושי הרמב''ם of Reb Chaim.

In the Ari שער הגלגולים we find that one's inner light comes from his mother and his outer light אור מקיף comes from his father. There is a lot to go into about this but I think I wrote about this a few times already.

In any case, this would not apply if the parents are on some kind of bad path. This is dealt with at length in the Tur Beit Yoseph