Not to toot my own horn, but when I wrote my gigantic overview for my ideas for a Season 6 of the 90s anime in the Fanfiction Ideas thread, I included my thoughts on how I'd have Usagi's family find out about her being Sailor Moon and how they'd react to finding out. The following is bit of both a summation and a fleshing out of those ideas.
I had Shingo be the first to find out since he was the victim-of-the-day at the time (and in a three-parter, no less). I imagined that he would of course be shocked that the heroine he admires and the stupid big sister he doesn't get along with were the same person all along. But, after seeing her save him, risking her life and such, he would probably come around to seeing past his sister's faults (but not completely ignoring them, of course) and recognizing her for the cool hero that she truly is. From this point, his relationship with Usagi would be more like that of Chibiusa's relationship with Usagi after Rubeus was defeated, in that Shingo would be more supportive of his sister being a superhero and could admire her more openly, but still tease her and such because, hey, they're still siblings who are bound to butt heads every now and then. He'd just be more willing to be nicer to her more often than he used to.
As for Ikuko and Kenji, I had then find out after their having been kidnapped by one of the main villains, who held them hostage to force Sailor Moon into a one-on-one duel with him (a duel that Sailor Moon ended up losing, but from which her parents were allowed to go free since the guy just wanted information out of Sailor Moon, info that he knew he'd only obtain in exchange for something that Sailor Moon would have wanted badly enough). Because of the circumstances of how they found out, I had it be that Ikuko and Kenji were anything but pleased to learn that their daughter was not only living a secret double life behind their backs for all this time, but that it was a life where she constantly threw herself into serious danger that could get her killed. The thought of losing Usagi to some murderous monster was too much for the two to bear and they insisted that Usagi end her career as a Sailor Soldier and go back to being only the normal girl that they thought she had been for all this time. Unfortunately, Usagi just couldn't give in to that ultimatum of theirs, which put her and her parents at odds with each other for a bit, not on good speaking terms either. Amid all of this, Shingo defending his sister for how cool it was to him that she was Sailor Moon.
It wasn't until another two-parter that saw Ikuko and Kenji (and Shingo) abducted again by a different set of villains, but this time taken along with other friends and acquaintances of the Sailor Soldiers: Motoki, Unazuki, Naru, Umino, Yuichiro, Rei's grandpa, and Hotaru's father. It took a combination of Ikuko and Kenji witnessing first hand their daughter successfully fighting to save their lives with her friends and teammates, and their hearing accounts from the other hostages (most prominently from Hotaru's father) about how Sailor Moon had saved all of their lives many times before and affected all of them for the better as a positive influence on each of them. Once they'd finally realize just how terrific their daughter has been ever since she became Sailor Moon, Ikuko and Kenji would finally come around to accepting her as the hero she has been toth to everyone else and to themselves. Of course, they'd still worry about their daughter always placing her life in jeopardy since, well, they are her parents and parents will always worry about their children out of love for them, but they wouldn't be as anxious about it anymore, having seen how capable a hero Usagi had grown into being.