I'll start off with the line itself:
The line seems simple enough. I know だけ has a few more meanings than just "only" but none of those appear to fit here either and I get the feeling "only" is close to what's intended, but not not quite right. Here my literal, word-for-word translation: "In only this situation that I can go out with him/her with parental authorization, I must think it a miracle." It's ugly but I think it makes sense, save for "in only this situation". It's plain my understanding of that is wrong. How should I be interpreting だけ here?