Can I ask a coworker about her dad's reaction to finding a rip in his pants?
Did his pants ripping make your coworkers dad upset?
She told me he is in a very bad mood. So I asked her what happened.
It's good to be open, within reason, with your coworkers at work about the events that are happening your life as it helps build a sense of camaraderie and friendship. As long as your coworker didn't have malicious intent behind talking with you about her father, it's something perfectly acceptable to disclose.
Her dad was looking at her weird, so I asked her what happened.
Well your coworker's dad doesn't work with you so he probably won't care if you ask your coworker about it. If they're willing to tell you, then there shouldn't be anything wrong and talking about your families could be healthy to make connections in the work setting. I say go for it.