As a young adult it makes sense that she should be left alone and free as long as she isn’t doing anything crazy in her parents home.
As a young adult living in her parent’s home, it makes sense that she respect house rules, understand that she is NOT free to do as she please, and if she doesn’t like house rules she should look for another place to live.
Parents have every right to establish rules in their own home for anyone who lives there-- child or adult. Usually parents let adult children have more freedom than a younger child, but may still ask an adult child to be home by a certain (reasonable and mutually agreed upon) time, not sleep over at a boyfriend’s house, go to mass, participate in family activities, etc.
So, no you view isn’t reasonable at all.
Growing pains are normal for young adults as they want more freedom and parents have to learn to let go more. It’s a negotiation. But some things aren’t negotiable, and I would venture that boyfriend sleepovers are non-negotiable.
Is it their business to expect her to tell them what she does outside of the house and try to stop her from going out?
Yes, it is. If she wants to live in their home, absolutely. If she wants to pay her own bills, not so much. Although parents will always be concerned about their children and their choices, and parents can be a great source of guidance as a person grows into adulthood. Your friend shouldn’t dismiss guidance out of hand just because they are her parents-- shocking, I know, but parents really do know things.
Or is that infringing on her freedom and privacy?
Nope.
She seems to think they are trying to impose their morality on her when she just wants to be left alone
It’s her parents’ job to raise her in the faith and to set good examples. It is also completely appropriate that they expect those living in their home to adhere to certain standards.
If she doesn’t like her deal, she’s free to leave. Then she can pay her own way and do whatever she wants under her own house rules.