Can you be approved for a British Passport/Citizenship if you don’t have your parents’ passport number/information?
She doesn't need any passport numbers. Apart from her own identification, she needs her father's British birth certificate and her parent's Marriage Certificate. She has those (they must be certified and not simply copies) so that's all she needs to prove her claim to British citizenship and can therefore apply directly for a British passport as she is already automatically a British citizen and has been since birth. As a British citizen 'by descent', she cannot pass her citizenship to you or your siblings unless you were born in the UK. https://www.gov.uk/apply-citizenship-british-parent/born-before-1983
You are assuming that everyone in the UK has a passport, well a huge amount of people do not. So it is not compulsory. As for the rest she needs to look it up.
You need proof of your parentage & proof of your parent's citizenship at the time you were born. Passport number alone is not proof, and not required. If your mother gets citizenship by descent, that does nothing for you. Citizenship by descent is only allowed for one generation & cannot be passed on.