Just shrinking the issue here ... its just if you have put name/details of both of your parents (which you should have) or only one? If you do not know then only IRCC can answer that. Its up to you weather to proceed further or not, with or without hiring immigration consultants. If you decide to proceed further to apply, either way, you need to include both of your parents to count towards family size, medical examinations etc (from your words, I am assuming here that your father and mother are still married to each other).I really appreciate your detailed response. Its very helpful for us to wrap our heads around this annoying problem from two years back. The challenge for us is that we dont recall which parents name we put, the mother or the father, we may not have put both or both. This is why we wanted to see how we can login to some site to find out which parent we put. And then have a letter explaining why we are including both parents in the actual application.
I have not seen any cases, in and out of this forum, neither I found any explanations for such cases on IRCC websites, so its just up to you to take a decision.
What I can say is, if I were you, I would have proceed further as I am okay to risk some 2500$ to immigrate my parents, given I waited for 7-8 years already and not sure how much I need to wait further.