You said she had fun shooting your sister's .22, right? (hint hint) You said you came on too strong perhaps, correct? (2nd hint hint) The "My Dad" comparison... probably a normal comparison in the overall scheme of relationships (maybe a 3rd hint, maybe not).
Any chance of you inviting "Dad" out to do some light plinking w/ a .22? Maybe getting a chance to do some (ahem) "bonding" with a chance to discuss "family values as related to self and family protection"?
Some people, maybe most, fear the unknown. Loud noises, explosions, etc can be off putting or downright scary to some. Some people will face that fear and attempt to conquer it. Others... may not. My GF, who holds a black belt in Aikido, owns a sweet little HK USC 9 but has "loud noise" issues (even w/ earplugs and muffs) and works hard on getting good groups, but her flinch is evident and she no smiles as a result. Put a Marlin 39 loaded w/ .22 shorts in her hands and she's Annie Oakley w/ a smile that lights me up. She does quite well w/ an AR15 but not so much shooting the Mausers...
You say she's a keeper.
Then...
Keep. It. Fun. (and always safety safety safety)
If that means .22's forever, nothing wrong with that at all. They come in all types and flavors, from cowboy plinkers up to Olympic quality target rifles. Cheap to feed, reactive targets, praise, smiles, FUN. Perhaps a class or two together. Maybe a monthly family range outing w/ dinner to follow... knitting by the fireside to close out the evening, perhaps.
As you can tell by all the responses, there is no single pat answer. Education is a start, safety a must, sublime responsibility and respect imperative... hopefully you two can grow old together, taking care of each others needs. Even if that means you agree to disagree on some things while always trying to find common ground on others.
If it becomes a control issue, that's another story you'll need to address at some point. Maybe. Maybe not.
But whadda I know?