Am I on the right path with this girl who has very strict parents (10 points)?

Well, I think you are in a good start Mike and you just used the word "like" instead of "love" so i guess your feeling is still on "getting to know her more" stage. Give more time, take it slowly. Just be sure of what you feel. When you are ready, when "like" becomes "love" already then just simply confess of what you feel. Nothing more, nothing less. If she would tell you to patiently wait like the junior guys, well, that is the best part. That is where the courting really begins and it will start whether you like it or not in there family, in there home Mike. Just be man enough to do it, 99.9% you will have not only her but her parents heart as well. Good luck and God bless.