We live overseas, but our mail here is unreliable. Can we use my parents' address in the States for our mailing address? I've indicated we live in a foreign location.

Yes. You can use your parent’s address as your mailing address. The IRS just needs a mailing address in case they have to send you something in the mail—or if you asked for a refund check instead of direct deposit.

The only issue you may have is if both your and/or your parents claim the Earned Income Tax Credit or Advanced Child Tax Credit. The IRS looks for instances where more than one household using the same address claims these credits as some families will split their households to illegally claim double credits.

Remind you parents to notify you promptly if you receive any mail from the IRS. Audit notices come with a clock and need to be addressed promptly.

The IRS might need the address only to contact you, BUT some states are aggressively looking for additional revenue. Be aware that some states are now going over federal returns looking for people with mailing addresses in their state who didn't file as a resident. With no other supporting data they are then sending out demand letters saying "we think you are a resident so refile your taxes and pay us $X plus interest and fees." I would highly avoid using any mailing address that isn't in the same state or country as your residence.

It's probably better to use your parent's address and if need be, deal with any letter from the state than possibly miss an important notice from the iRS.  Some IRS notices give you only a limited time to respond.