As a person that has suffered from depression for all of my adult life, I'd like to share one piece of advice. when one is planning to do something that is part of getting the help needed, you cannot let anything stop you. While completing your parents assigned chores is top priority, top top priority is you getting help. In the future, tell them that you'll fulfill their request AFTER you've completed the necessary steps to take care of yourself.

Doing something actually helps with depression.

Just go, forget the chores. get yourself looked after first. No punishment for not doing chores is worse than your mental health.

just go and when you get home get them done, or if you have some time during the day before your appointment then do one or two chores, take a mental health break for 10 min, then do one or two more, then have another break, and finish up if you can before going to therapy, also let your therapist know that you struggle with that many chores! I'm a stay at home mom with ADD and depression so both of those things make it very difficult to stay on task and actually get simple things done. most days I have to mentally prepare myself to get these things done. judt try some different things to help make it easier for you, I know all this shit you constantly have to do completely sucks but it's also part of life as well and you will never NOT have to do it, but you can find balance to make it not so overwhelming leaving you feeling exhausted.

Mental health is just as important as physical. I wish you make them understand it and take care, OP.

don't tell them when your appointments are or lie and tell them a different day all together (tell them it's on Monday when it's really Weds then when they load you up on Monday you'll be free to make your appointment)