If I am not being claimed as a dependent, why are they asking for my father's information on my taxes? Does unemployment count as child unearned income?

Yes, Unemployment compensation is considered unearned income. 

if you received unemployment compensation you may be subject to the kiddie tax.

The Kiddie Tax applies to all children who are 18 years of age or under—or dependent full-time students between the ages of 19 and 24. The kiddie tax applies to most unearned income that a child receives and does not apply to any salary or wages.

All unearned income over the threshold is taxed at the parent’s marginal income tax rate rather than the lower child’s tax rate.

In order to calculate the tax at your parents' rate, TurboTax needs information from your parents' tax return.