I and my spouse want to file "married filing jointly", however her parents already filed their taxes (on paper) and claimed her as a dependent? Do we have to need to file extension and wait for her parents' amended tax return (1040X) to be processed, or can we file our taxes on paper right after they submit their 1040X?
You don't need to wait until they file their amended return. You can file your jointly filed return now, but you will need to mail it in. You won't be able to e-file if she has already been claimed elsewhere.
Amended returns take a long time to process-up to 16 weeks. You can file an extension if you wish, but you will need to pay whatever you owe before the due date of the return.