In my opinion, it should work. However, it will be good if you attach an extra document like passport, which would corroborate your DOB and your parents name.
Visit your Birth Registration office and apply for a new birth certificate. It will not take them much time to issue a new one. Even i had the same problem and i got it. Note: Make sure you have full name of yours and your parents in the birth certificate.
EE doesn't ask for it but some provinces do. OOPNP specifically asks for a copy of your birth certificate or an acceptable national document where
an original birth certificate is not available (i.e., letter from appropriate government authority indicating why your birth certificate is not available).