If I understand you correctly, your logic goes wrong right at the beginning. It sounds like you think one could avoid the income tax that would otherwise be owed to the US because of earning the money that was sent as a gift.
That's not normally true. From the IRS's Gift Tax FAQ:
May I deduct gifts on my income tax return?
Making a gift or leaving your estate to your heirs does not ordinarily affect your federal income tax. You cannot deduct the value of gifts you make (other than gifts that are deductible charitable contributions).
So the person who sends $10k to their parents doesn't pay any less income tax than if they had kept the $10k in the US, or had just send the $10k overseas directly to their own bank account. Gifting and re-gifting didn't accomplish anything from the point of view of IRS taxes.
You may have been confused by the "annual exclusion" that's mentioned on that same page. This exclusion is an exclusion for the gift tax. This is a separate tax on gifts, usually paid by the person who gives the gift. If it weren't for the exclusion, one would pay taxes twice on the money sent to their parents: first, when the money is earned, and then again when the gift is given. The exclusion helps avoid this second tax.