i'm currently in the process of moving from Canada to the UK. Let me give you some advice: do not go on vacation internationally with your bird.
Conures, like all parrots, are classified under Appendix II of the CITES exotic animals list. This basaically classifies them as extremely endangered and international movement is restricted.
Your bird would not only need LOTS of paperwork and permits, some of which cost hundreds of dollard, s/he would also need vet visits and quarantine in every country before you cross the border (except leaving Canada, although Canada requires a minimum month long quarantine on return to Canada).
You have to pay those vet fees, and you also have to pay the quarantine fees.It's very expensive.
It's also very stressful for your bird. They are sensitive animals and need a structured routine. Think of all the stressful factors this entails for your bird:
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being kept in a car for extended periods of time (akin to being caged for all that time, since s/he won't have a playstand/ toys/ ability to stretch her wings)
-repeated vet visits, examinations, sometimes bloodwork
-changing temperatures all the time as you're driving through different lattitudes
-changing timezones which results in a schedule being thrown off
-driving down through the longitudes therefore entering zones where the time of sunset/ sunrise is constantly changing much more rapidly than normal for a bird
-quarantine periods- enough said
now consider how stressful this will be for you:
- constantly having to worry about finding accomodations that will let you bring a bird in the room
- worrying about your bird getting sick (likely, with all the change to routine) and having to leave her behind at the border because she can't pass a vet check. The usual policy here, especially in south american countries where conures are native, is 'don't worry we will find a home for her' before she finds herself launched out of an open window
- the cost- it will cost you an absolute fortune
trust me- it's not worth it. Find a reliable foster home, bording aviary or friend or family member to look after her if you have to go.