Gina, I'm wondering: when is it ever going to be okay with your mom for you to transition?
You've already waited 20 years. Twenty years since they found out you're trans. You've tried the medication, you've tried therapists, you've done everything they've asked of you. I'd call that giving it a good ol' college try... and you're still trans.
So right now your mom wants you to hang on so that you can support your father. But in the very sad event of his passing, do you honestly think she'll suddenly say "OK Gina, thanks for helping with your dad, I'm happy for you to transition now?". Or is she going to move the goalposts again? What'll be next - wait until your mother dies? Wait until your sister's kids grow up? Wait until the second coming?
Gina, honey... I honestly believe she's trying to stall you in the hopes that she can put you off transitioning in her lifetime, and preferably forever. You've mentioned that she's very controlling and manipulative - and I believe that's exactly what she's doing to you now. But here's the thing: she can only control you and manipulate you if you allow her. If you want to transition, you'll need to break the cycle.
Here's what I would do: tell your mom that you're attending your therapist appointments on your own - you're an adult and are entitled to privacy. Either call your therapist out on his 'man-up' comment, or drop him like a hot potato. And tell your mother that you'll be more than happy to do your share of helping your father - which is to offer 1/4 of the help he needs, since he has three other women who can help him too. But tell her you won't be doing the lion's share because you're going to need a lot of time, support and money to successfully complete your transition. Your mother and sisters need to 'man up' too.