I'm not sure that this will help you, as I'm not sure why they're asking for nationality at check-in...and yours may be different from your parents'. Typically I thought it was just passport information. So perhaps don't take this as advice and ask WizzAir instead so you don't have problems at boarding.

That said, "nationality" usually refers to your country of birth.

For example, I have a friend who was born in England, but became a US citizen several years ago. His nationality is British, but his citizenship is US.