With the choice in front of her, Saki was put in a bit of a dilemma. From what I could see in her eyes, she definitely wanted to do what I suggested. There was even a look of anticipation in it.
Introducing me to her parents was probably one of the things she wished she could do as well. There’s a reason why she couldn’t suggest it openly. If I added what she just told me about them being aware that she already got herself a boyfriend and they’re not welcome of that though, she’s probably afraid that bringing me in front of them would result in a negative outcome.
Her confidence that her parents would accept our relationship was low. What's more, she also has to lie to them about the true nature of our complex relationship. Even if we can hide it initially, there will come a time when we will have to come clean in front of them.