This summary is not the rule. The rule text controls.
If you believe you can't get a fair trial in the parish where your case is being heard, you can ask the court to move it somewhere else. This is called a change of venue. To do this, you file a formal request called a contradictory motion, which means the other side gets notice and a chance to respond before the judge decides.
You'll need to give the court a real reason why the trial won't be fair. This could be because the other party has too much influence in that area, because people in the community already have strong opinions about the case, or for some other good reason that convinces the judge.
If the judge agrees and grants your request, the case gets moved to a different parish. That new parish must be one where none of the parties involved in the lawsuit actually live. This keeps the trial location neutral, so no one has a home-field advantage.