My boyfriend’s parents invited me on holiday with them. How do I ask permission from my strict father?

YOU shouldn’t be the one doing the asking, sweetie. Since your boyfriend’s parents issued the invitation to you, THEY are the ones who need to take the initiative and ask YOUR parents if you have their leave to go on the trip. If they can’t or don’t want to do this, then I’d think twice before saying anything at ALL to my parents, if I were you.

Your bf’s parents asked you to come along, which means that they must be willing and able to monitor your behavior on the trip, not to mention being responsible for your safety. YOU need to show your parents that you are responsible enough to be allowed to go- that means you are polite, get your work done when you should, you don’t talk back or argue when asked to do something, and you do your fair share of the chores. The other big reason why your bf’s parents need to do the asking involves the issue of costs of the trip. Both his parents and yours need to decide who is paying for what in terms of airfare, hotels, ground transport, food, sightseeing costs, taxes, transfers to and from the airports in each place you visit, any and all tips and gratuities, and so forth. ( Even if you sail to one or both places you have mentioned, you will still probably have to fly home, and that, along with hotel costs, will be your biggest expense of the trip.) Your parents need to discuss ground rules for your behavior with your boyfriends parents as well- and one thing that needs to be clearly settled before anyone does anything is what sleeping arrangements you will use and be allowed to have. This brings up the issue of sexuality- and since you have already said your dad is strict, I strongly suspect he will only agree to the trip if it can be made clear that you are not going to come home from it pregnant or with an STD. That may mean you and your boyfriend are going to have to sleep in separate bedrooms- are you going to be able to handle that all summer?
Traveling is a wonderful experience while you are young- and it’s good for you to see as much of the world as you can and get to know other cultures and nations. I think that if your boyfriend’s parents are really serious about the trip, and are willing and able to reassure your dad that you will be safe and not get into any kind of trouble while you are abroad, you should be allowed to go. But again, YOU are not the one who can ask for permission to do this- nor should you try to. Let them do the talking, sweetie- they are the ones who made the invitation to you in the first place, so they should be the ones to follow up on it. Good luck- this sounds like a great experience for you.