According to HealthKnowledge, the main disadvantage of parametric tests of significance is that the data must be normally distributed. The main advantage of parametric tests is that they provide information about the population in terms of parameters and confidence intervals. Another advantage of parametric tests is that they are easier to use in modeling (such as meta-regressions) than are non-parametric tests.
Surender Komera writes that other disadvantages of parametric tests include the fact that they are not valid on very small data sets; the requirement that the populations under study have the same variance; and the need for the variables being tested to at least be measured in an interval scale.
Chong-Ho Yu states that one rarely considered advantage of parametric tests is that they don't require the data to be converted to a rank-order format. Conversion to a rank-order format in order to apply a non-parametric test causes a loss of precision. Another advantage is that it is much easier to find software to calculate them than it is for non-parametric tests. Also, unlike parametric tests, non-parametric tests only test whether distributions are significantly different; they are not capable of testing focused questions about means, variance or shapes of distributions.