Blingy has given you good advice.
You say that youíre going to a country with a completely different climate. Looking at the information above your posts, Iím assuming that youíre moving either to or from Canada. When you say Ďmight be two ways tripí Iím guessing you mean that it is a long haul flight, possibly split with a transit stop.
Is this country move via airplane?
Letís try to be practical here - Iím not sure whether youíre a young adult or a older child but whilst your intentions are very honourable, you are still dependent on your parents. You are finding it difficult to purchase basic resources for these birds. Do you feel that your parents are willing to complete the relevant import paperwork (possibly with vet health checks) and pay additional flight fees for your budgies?
Importing or travelling with pet birds from countries other than the United States - Canadian Food Inspection Agency
Whatever the journey is, I suggest that you try to find a time today when your parents seem relaxed and have a practical discussion as to the future for these birds. Your parents will have many things to be organising at present. If they are not prepared to look into how these birds can be taken with you, then my advice is to just accept that.
I understand how disappointed you will feel, but try to keep the harmony in the family and transfer your energy into finding an alternative home for these two.
If you are on Facebook, maybe you could post an advertisement for your friends to see. You could place a printed poster up at work or school (if at school, ensure that any replies have parental agreement). Look up animal rescue/rehoming/adoption sites to see if they will take them. If all else fails, try to return them to a pet shop and perhaps ask the owner if they will take them in, as Ďavailable free providing a larger cage is bought from the storeí. Whatever you chose, you could type up their background story, what successes youíve noticed so far and also provide a link to this website for further advice, support and research. Reading the birdsí story, helps readers make an emotional connection with them and increases your chance of success.
I would be delighted if you could manage to bring these birds along with you, but to let the next 3-4 weeks slip by on just a futile hope, isnít the best plan.