Mike ~
Obviously, you should not allow your parents to have so much influence over you when you're already stressed with your AN situation but, as you stated, this family dynamic goes back 45 years so it won't change overnight, if ever, considering your parents advanced ages.
I wouldn't hide the information about your re-growing AN, that would be incongruous in this situation, but I would tell either parent only what you feel is necessary and if the questioning becomes intrusive and/or unsettling, calmly but firmly state that you're quite able to handle the situation and that you don't need advice because you have a doctor and are in contact with the ANA. Then tell them (again, gently but firmly) that you have no intention of talking about it any further. Of course, your parents may not accept that and insist on talking about it (and offering their opinion) in which case you can simply chose not to respond and if they demand a response, again state that you have nothing else to say because you're taking care of it. If you do this with a cordial attitude, they shouldn't be offended - but frankly, we cannot control other people's reactions and if they act offended, so be it. I'm sure they'll get over it soon enough.
These adult child-aged parent issues are thorny and I don't presume to have any expertise in dealing with them (my parents are deceased) but I believe that visiting your parents (in Florida) and telling them, as you should, about your AN issue shouldn't be a struggle if you decide ahead of time to take control of the situation and not be unduly influenced by their reactions. As I stated, I know it's a thorny issue but you did seek advice and, for what it's worth, this is mine. I hope it helps a bit.
Jim