Harshipping is being able to register a horse with unregistered/unknown parents. For example, if you have an appaloosa colored pony under 14h, you can hardship it into POAC. If you have a mini under 34" you can *until 2013* register it with AMHA (for extra fees of course... a mini stallion is an extra $1500, for example).
AHR does not do hardshipping, and the foal's sire must be licensed as a breeding stallion with AHR in order for it to be registered.