The two main advantages to using parametric tests are greater precision and power. Field (2012), however, notes that there are some situations where nonparametric tests have greater power than parametric ones. The advent of permutations and randomisation tests have also enabled confidence intervals to be computed for test statistics based upon distribution-free tests.
(The below is taken from Alex Yu's reply to the edstat discussion list on here which discussed advantages of parametric tests).
Edgington (1995, p.85-86) ‘it is unwise to degrade the precision by transforming the measurements into ranks for conducting a statistical test’. In addition, generally speaking, the statistical power of non-parametric tests are lower than that of their parametric counterpart except on a few occasions (Hodges & Lehmann, 1956; Tanizaki, 1997).
Further, non-parametric tests are criticized for being incapable of answering the focused question. For example, the WMW procedure tests whether the two distributions are different in some way but does not show how they differ in mean, variance, or shape. Based on this limitation, Johnson (1995) preferred robust procedures and data transformation to non-parametric tests.
Fagerland and Sandvik (2009) suggests using the Welch-Satterthwaite correction to the t-test instead of the Mann-Whitney test if the two groups being compared differ on standard deviations and skewnesses. Fagerland and Sandvik (2009) and Zimmerman (2003) also found in addition to Mann-Whitney giving inflated Type I errors when groups have unequal standard deviations it can also give misleading results when comparing large groups.
Edgington, E. S. (1995). Randomization tests. Third Edition. Marcel Dekker Inc:New York.
Fagerland, M. W. and Sandvik, L. (2009). The Wilcoxon-Mann-Whitney test under scrutiny. Statistics in Medicine, 28, 1487-1497. A pdf copy is here.
Hodges, J. L. and Lehmann, E. L. (1956). "The efficiency of some nonparametric competitors of the t-test". Annals of Mathematical Statistics, 27(2), 324–335.
Johnson, D. H. (1995). Statistical Sirens: The Allure of Nonparametrics. Ecology, 76, 1998–2000.
Tanizaki, H. (1997). "Power Comparison of Nonparametric Tests: Small Sample Properties from Monte-Carlo Experiments," Journal of Applied Statistics, 24(5), 603-632.
Zimmerman, D. W. (2003). A warning about the large sample Wilcoxon-Mann-Whitney test. Understanding Statistics, 2(4), 267-280. A pdf copy is here.