Parents & Children, could there be Rivalry?

Have you watched "proof" starring Antony Hopkins and Gwyneth Paltrow? I just did, and a key question came to me afterwards. How do parents & children c each other's achievements? What if those achievements are of a competitive nature? What if one's success highlights or implies the other's failure? What if the one who is failing is the one desperate for success?

If those question's seem awkward watch the movie and you will know what i mean - Hopkins and Paltrow are doing a great performance.