Can I import custom fields with my contact list?

You can create new custom fields (i.e. Parent's name, Parent's phone, Quiz Score, etc.) to the Contact's Profile information (Website > Forms > Profile Form > Extended Fields Section).

Once you have created those new fields, you can then add that information to the Contact List you wish to import (making sure the column name matches the field name you created). 

When you import your Contact List those fields (and information) will also import as well.