Does my seeking forgiveness for one of my parents for a specific sin they committed suffice as their own seeking of forgiveness, knowing that I do not know if they seek forgiveness or not, and I am too embarrassed to ask them?

Praying for one's parents and seeking forgiveness for them is a prescribed and beneficial act for them both in their lives and after their death. However, it does not nullify their obligation to repent, as repentance is an individual obligation upon a servant. Your parents must initiate repentance, and you should remind them of this in a respectful manner without mentioning their sins.