Scoring question - Earned runs

[BratzCoach]

I was always under the impression that if a runner scored due to an error, then it was an unearned run. However, I had a debate with another parent during a game the other day as to whether or not the run could become an earned run later.

Here is what happened:
There was one out, and no one on base. The batter hit a ball to center field for a single. The fielder misplayed the ball, and the runner circled the bases to score. This is an unearned run, since the batter would not have scored without the error by the centerfielder.

The next batter hit a home run that cleared the fence. The parent next to me said that now the previous run is earned, since the runner would have scored on the home run. I--and the home plate umpire whom we made the inquiry to--said that once the run is unearned, it's unearned.

Any answers?