The family-wise error rate is not affected by whether the test is parametric or not. It is a product of the alpha level chosen for the hypothesis testes and the number of tests used and the nature of the decisions made on the basis of the test results.

You don't describe your data or your problem, so I can't give specific advice, but do you actually need multiple comparisons? How many hypotheses are you testing? Do you need to use a Neyman-Pearson approach of automatic decision rule-based inductive behaviour? Perhaps you could characterise any differences and make inference on the basis of their magnitudes and the scientific consideration of what they are and how they were observed.

I think that some permutations tests (those that are not based on ranks) have higher power when the variances of the groups are similar, so you might need to transform the data before analysis for best results.