The standard for defamation in the case of public figures (for which the parents of the Sandy Hook victims probably qualify because it's that broad) is "actual malice"--the speaker needs to either know that the statement was false (at the time it was made) or entertained serious doubts as to its veracity. For non-public figures, it's generally lowered to "reckless negligence". The other required element for defamation is that it has to be damaging--you need to show that you suffered as a result of the allegedly defamatory statement.

If it's at the point where the statement is causing people to issue death threats against you, it's not a "potentially genuine mistake".