Some countries did not have government-issued birth certificates until quite recently. Evidently India is one of those countries.
My husband is from Nigeria, where people born in the 1970s and earlier will almost certainly not have a birth certificate. Babies were born at home, the government didn't care, there is no record. My husband said people in remote villages sometimes don't even know how old they are. However, they can go to school, get passports, etc. My husband got his passport by way of an affidavit. His mother went to court and told them when her son was born.
So, when I was filling out the forms for my husband's PR application, I also wondered what to do about the lack of a birth certificate. I included a photocopy of his baptismal certificate, his passport of course, and an affidavit from his senior brother (now the head of the family because both of my husband's parents are dead) stating my husband's birth date.
I phoned CIC to ask about this; they said to include as much proof as I could get, and that what I had was fine.
The visa office that processed his application had no problem with the lack of a birth certificate. Visa officers working in countries where not having a birth certificate is common will know this, and not be surprised or suspicious.
For the OP: what you have sounds good, but I'd get an affidavit (from the parents if possible) as well. Also include a letter explaining the circumstances. This has never been a problem for my husband, so you should be OK too. Also phone CIC and see if they have any advice.