Parametric tests are more robust than non-parametric tests. This means that they are more powerful - they are more likely to correctly identify a significant difference when there is one, and therefore the greater power they have in correctly rejecting the null hypothesis This means there is less likelihood of a type one or type two error.
Parametric tests have more power than non-parametric tests because they have more information about the data and populations from which the sample is drawn (normally distributed sample and an interval (at least) level of measurement for the data). Also more sophisticated mathematical calculations can be done with interval level data. I hope this makes sense.
Another criteria is homogeneity of variance, but I have never really bothered explaining this to students in any detail unless they ask - it just means the standard deviations of the 2 sets of data are similar.