It's down to narrative distance. If you want your third-person narrator to feel more distant or external to the pov character (assuming the pov character is the person whose parents it is), then "her mother" and "her father" work. If you're going for a deeper limited third pov, one that feels more like first without the "I" pronouns, then having them referred to as "Mom" or "Dad," or however she thinks of them, in the narrative is probably the best approach. It might depend on the perspective of the pov character at the time too. For instance, I might refer to my mom (as I just did) as "my mom," when being a bit more formal, as when I'm mentioning her to someone who doesn't know me or my family well, but I think of her as "mom" and may refer to her that way when talking about her to someone in my inner circle.
If the story or scene is from the pov of someone else who would think of them as "her mom" or "her dad," of course, then that would work in a deeper limited third pov too.
This means (in multiple pov books) that characters may be referred to differently in different scenes, depending on whose pov the scene is written from. For instance, referring to a teacher from a kid's pov, she might be "Mrs. Smith," but in the pov of a friend, family member, or colleague of that teacher, she might be "Mary." From the pov of the kid's parent, she might be "Billy's teacher."
Referring to parents as Mr. or Mrs. works if you're in the pov of someone who would think of them that way, or are in an omniscient pov in a story aimed at kids (where adults are generally referred to with salutations.
[Edit--CF cross posted with me] As for "Mrs." being offensive, that's in the eye of the beholder. Many married women who use their spouse's surname still think of themselves that way, or only use "Ms." or "Dr." or whatever in professional settings. It's so ingrained to call a mature or married woman "Mrs." still that those of us who didn't change our names in marriage (or otherwise don't feel right using "Mrs.") generally just let it go when people call us that. I do correct when they call me by my husband's last name, though (or when someone calls him by mine).
The main thing is, the use of salutations generally conveys a formal tone, and if you want your pov character (or external narrator) to feel formal, it can work.