But Parler is not moderating at all

The irony is that Parler is highly moderated. I haven't tried it myself, but I've heard it's very easy to get permanently banned like immediately.

The problem isn't who's moderating more than whom. The problem is who's getting moderated more than whom.

Twitter moderates conservatives quite heavily and leaves liberals (like Kathy Griffin) more freedom. Parler does exactly the opposite. It heavily moderates or bans left-wing people but leaves conservatives mostly free.

Depending on how far you want to go down the conspiracy rabbit hole, you can argue that the issue is then not that Twitter is more fair than Parler, but Twitter is just more media-friendly (read "liberal-friendly") than Parler and fairness has nothing to do with it.