So I plan to use some miles for a award ticket for my parents and I, but from what I gathered is that miles can only be used for the account holder and immediate family.

So for me there is no issue and for my father, but I was wondering how strict are they in verifying the names?

My mother has both her maiden name and married name on the passport.

For instance:
Me: Jason Smith
Dad: Bob Smith
Mom: Mary Doe-Smith

Is that allowed? Would something like that cause issues?