Sie erlauben es ihren Kindern, mit Schmutz zu spielen.

This is currently translated “They authorize their children to play with dirt.” – “authorize” seems awful stuffy, formal, and unnatural for the context, especially as “erlauben” is usually translated “allow” and other words like “genehmigen” are usually used for more formal words like “authorize”.

I think it would be better to just say “allow” here.