My [29M] parents [60F/61M] want me to reach out to my no contact sister [27S] and it blew up yesterday...

Yesterday, I [29M] met my parents [60F/61M] for a hike with my wife [28F]. As we were hiking, my dad and I got a bit ahead of my mom and wife and he brought up a topic we've been circling for a few weeks - they want me to reach out to my estranged sister for them.
My sister [27S] has been estranged from our family for over a year. My sister has suffered from severe depression since we were teenagers. My parents not having any previous experience with depression did, in my opinion, the best they could. They took her to therapy, tried anti-depressants, and my mom was especially always a listening ear. Were they perfect? Definitely not. But they made an conscience effort to help her get to a better place. Since her teenage years their relationship has always been rocky.
Personally, it wasn't until I went to college and had a suicidal roommate that I really began to develop a true understanding/empathy toward mental illness. When we were kids my sister could be very aggressive and cruel towards me so I began distancing myself as we got older. Our relationship was never great, but was really bad throughout middle/high school. It wasn't until we had both graduated college that we really started talking and attempting to foster a relationship.
Our early adult relationship went in ebbs and flows of talking. To be frank, it's always been a bit of walking on eggshells with her. We'd be having a pleasant conversation, but after 15 minutes it'd go sour really quick. I was never sure what I could say or talk about with her even though I'd try to establish triggers and topics we should avoid. A few years ago she started opening up to me about our childhood and wanted me to confirm that our parents were abusive people. I told her I couldn't agree with that experience and that I had a great relationship with mom and dad. I told her though that I understand we may have had different experiences of the same situations and I would listen to her and be there to talk if she need. For the next few years, we'd have the occasional conversation about, "Do you think mom and dad are abusive now?" "What about this situation?"
It all exploded after a family vacation to my uncle's ranch house. We all agreed to bring food, had rented a boat at a nearby lake, etc. She didn't bring food, slept in extremely late every day and make us late to scheduled activities, was very cruel to her boyfriend and his family in front of us, and picked fights with almost everyone in attendance. After we all went home my mom called her and let her know that her behavior was very frustrating for all of us and to "see what her deal" was. You can guess how that conversation went. My sister called me in tears, begging me to admit that they were abusive and terrible parents. I told her again that I couldn't agree with, but respect her emotions/ her different experiences and was there for her to talk/listen it out. We went back and forth for awhile, but eventually I suggested that she just take a breather and give herself some time to cool off and heal. Maybe a month or two of minimal or no contact with them to figure out a plan and talk with them. She said she'd sleep on it. The next day I get a text that she doesn't want to talk with me any more and she regrets our conversation. My parents got something similar.
Over the course of the last year-ish I've had to deal with daily and weekly phone calls about her with my parents. I finally got my parents to go therapy (thank god) and I felt everything was moving in the right direction. While I wish I could be there for my sister in a more active way (without having to condemn my parents), I really hope this time for her has been healing and that she's finding a healthier way to live - even if it doesn't include us.
So back to the hike. My dad had been hinting that he wants me to try and contact her for a few weeks, but he thought our marsh hike would be the appropriate time to bring it up. I tell him that I don't think it's a good idea to reach out and force a conversation. We know from two older cousins of mine that my sister is still alive and doing her own thing. He starts getting a bit loud and talking over me, and I ask him to slow down and listen to me for a minute. I say, she asked us to not contact her so I want to respect her wishes and that if she does decide she wants to talk with us I want her to feel like she can trust me. Part of that trust in my opinion is respecting her wishes. So I say I won't do it. My dad lost it like I've never seen him act before - I was completely in shock. He started saying "We're *&%$ing dying here and you won't help us!" (I guess he forgot about the last year...) "I'd do anything for family and you won't try to talk to her for 60 seconds for us?" Etc. I told him again I don't think it's the right thing to do. I also said what's the plan if she answers the phone? Do I tell her that we miss her and want to talk? Are you ready to apologize to her? Will you fight her and drive her farther away? What's the plan? Of course, he didn't have any answers to these questions and continued to rant. I told him that I couldn't talk to him while he was like this and if he wanted to talk like adults he needed to calm down. I want to help them get through this, but this isn't helpful for us or her. I started back down the trail toward the parking lot and bumped into my wife and mother. My mother went to console my dad and my wife and I went a separate way and talked.
We all met in the parking lot at the end and he gave me a half-apology/continued talking about it. Tears were shed, more angry comments for said, and eventually I just stopped talking so we all left. My parents have both since texted me apologies and that they feel sick about what happened and to contact them back. It's the first "real" fight I've ever really had with them so I'm taking some time to meditate on the situation and come back with a clear head. I know they're in deep pain and at the end of their rope with this, but it doesn't excuse my dad's behavior toward me. I can forgive them because I know I have no understanding of the pain of losing a child, but I have to make sure that if I continue to be there for them I'm not treated poorly or breaking my sister's boundaries.
How can I[29M] help my parents [60F/61M] navigate having an estranged child [27F] and get them the help they need before our relationship hits a hard place? How can I take care of myself in this situation better?
TL;DR: I [29M} have an estranged sister [27F]. There is a lot of history and difficult issues. My parents [60F/61M] want me to contact her for them, but I said no because I want to respect her wishes. My dad blew up at me and I'm trying to understand how to proceed with our relationship/family troubles.


Source.