Yes - not only is this allowed, Apple and AT&T have systems in place to effect this purchase. Note - this is only for plans where you are billed together. If you can get on the same plan, just ask them about using a cross upgrade so that you can burn the upgrade eligibility of a different number on your family plan to get a new phone for a line that currently isn't eligible for upgrade pricing.
In your case, where you are not billed together it gets messy. If they don't buy the phone off contract - then the carrier could hold them to pay for the data plan if the contract has that provision. This situation is analagous to them buying a new phone and then selling it on the open market. I haven't heard of AT&T cracking down on this, but you could easily see how they intend the subsidy to generate 2 years of revenue from the buyer.
As you are not eligible for an upgrade, it goes to sense that you haven't yet paid off your last "loan". Have you asked how much it would cost for you to end your contract? Unless you also are on a family plan, you might shop around and then go to AT&T and not threaten to leave, but explain your budget and your options and ask them how long you have to wait or how much you might need to pay to get back to a more budget friendly place for you to upgrade your handset. I've seen times when they'll budge a few months, but keep in mind if you haven't honored your last contract with them, you've got to be reasonable when asking for further accomodations.