What recourse do I have regarding an emotional support animal in a no-pet condo?

This is problematic case because you and your doctor are fostering the very sham that causes people with legitimate ESA/SA needs so many problems. This is clearly an attempt to abuse the ADA to allow you to have a pet where they are prohibited with a fake claim of ESA status and need. That you magically have medical need for an ESA at the same time your parents pet, may have to go to shelter, that you cared for as a pet, is magically your new ESA. Your Dr. claimning it would violate HIPAA which is not the issue is likely signaling that they will not risk thier license or being sued for fraud by backing the bogus ESA letter further. The option here is accept that they called our bluff or double down on it and spend a good bit ofmoney on a tenant lawyer to push the ADA/ESA issue to get concessions or to sue them for illegal discriminaton hoping that they will capitualte before deposing your doctor or the doctor will back the issue properly to avoid exposure. You will need to decide if its worth spending thousands of dollars right out of the gate for such an effort.