|How to Start:||Talk to Parvati|
Travel to Groundbreaker and as you go to leave your ship you should find Parvati waiting for you near the exit. Talk to her and she'll request you take her to meet Junlei Tennyson, the Groundbreaker's chief engineer. Parvati's hoping she can learn a thing or two about starship maintenance, and since it's purely beneficial for you on multiple levels, agree.
You'll find Junlei at the northern end of the Promenade, in Engineering. Head inside and take the first right (east) to find her office. Chat with Junlei and, after she flexes her authority and some small talk, urge Parvati to chat with her (repeatedly) and after suffering through a painful bit of dialogue you'll be free to return to your ship.
Return to the Unreliable and talk to Parvati who will subject you to a poem she got from Junlei. When you get the option to say "Sounds like she likes you.", do so, despite the fact that it leads to learning more about Parvati that you possibly care to, after which you'll be given a meaningless choice in response to revelation. Another romantic crisis will immediately follow, as Junlei's next message mentions... another girl! What is Parvati, a jealous teenager?
Apparently so, and this childish romance plot takes another tedious turn when Parvati asks you to take her to The Lost Hope on the Groundbreaker so she can get drunk. Well, given how incompetent Parvati is at simple interpersonal interactions with other adults, you'd be wise not to allow her to try to get drunk at a seedy bar on Groundbreaker by herself. Leave the Unreliable and return to the Groundbreaker, ensuring Parvati is in your party, then head on over to The Lost Hope.
Talk to Parvati when you get inside the bar and make whatever alcohol suggestions you please and after one swallow she'll start whining about Junlei again. Try not to be too mean, and after she says "I- I need another drink. Right now. Afore I lose my nerve." Respond with the following:
- [Persuade 25] Let's get you some water.
- About what?
- There's an easy way to get answers to all those questions.
- You can't control how people react to you, even people you care about.
- The only think you can control is you.
- Something to work on, then.
- Are you going to ask her out?
- [Persuade 15] You're still unsure?! Yes! Just ask her! or [Determination 15]/[Inspiration 15] Don't ask for my opinion. Believe in yourself.
This is just one of many ways to advance through this conversation, which... probably doesn't matter in the long run, but hey, extra dialogue options at the end with skill checks has to mean you've done something right... right? Whatever the case, after she's had enough of your company, or has come to some decision regarding Junlei, she'll ask to go back to the shit. Err... ship. Close enough.
Return to the Unreliable and track down Parvati and she'll tell you about her latest correspondence with Junlei, which will finally end this quest. If you followed the instructions above, it will have led to... healthier relationship practices on Parvati's end, which will result in the follow-up quest Don't Bite the Sun, which should start the next time you leave the Unreliable and return and talk to Parvati (at any rate we were able to start this quest by transitioning between the Groundbreaker and Unreliable).