Took the flu shot for the first time this year. Should my 78 year old mother take it as well?

My doctor recommended the influenza vaccine now that I am diabetic, since infections apparently affect sugar control. I took it last week, had a sore arm for a day, otherwise no issue. My mother is 78 and lives with us. She has BP and mild asthma. She is against injections in general and says she has managed all her life without a flu shot. My wife thinks we should insist. For those who have elderly parents at home, did you get them vaccinated? Was there any reaction? I want to go to her doctor with a proper argument and not just "the internet said so".