The laws aren't here to cater to the individual but to advance the greater common good.
This would happen in an ideal world, where all laws would be created through an evolutionary process that already converged to an equilibrium. New laws voted by legislators frequently fail to pass simple common sense tests, so they aren't always good for the community.
If we accept that the current laws regarding minimum age for civil responsability are adequatedly fine tuned by evolution, it is correct to deny permission for a minor to go around the world alone.
To justify breaking this law, one has to believe that the minimum age should be adjusted to 13 years old as a consequence of new circumstances introduced. Although this might be the case with many laws, it is not true on this one, for the minimum age for civil responsibility has been increasing in the last 5000 years.
Before biblical times, one was deemed responsible for his acts when able to diferentiate the left hand from the right one (4 y.o.). Then we started to use puberty (13 y.o.), which is easily verifiable and makes sense for reproduction.
More recently, since the total volume of knowledge increased due to science and press, the age increased to 16 to 21 years old, more or less coinciding with the age when formal education is completed. This timing is reasonable because one can not obey laws that he can not understand, so learning should precede responsability.
Since the knowledge database, the technologies available and the minimum required level of personal sophistication are in expansion, and since people have more access to education, it is quite obvious that the minimum age tends to increase and not decrease. The existing law is complacent, if anything.
The girl's parents accepted to expose her to risks and responsabilities that were acceptable at her age in the past, but not nowadays. If they fail to understand this situation, the state's intervention is justified.
On the other hand, if my daughter wanted to do the same and I knew she was up to the task (which she isn't), I would move to a country in an earlier development stage, where the law would be older and the minimum age lower. Then all could be done legally. And I would have to accept that my education and sophistication was in its "early development stages" too...
