I filed a paper extension (not through Turbotax) on April 15 with zero tax due. How do I record this in Turbotax so it shows that the tax return was extended?

There is no place to record that.   If you sent an extension request by mail then the IRS knows you have an extension.    If no tax was due, it is a moot point anyway because there is no penalty for filing late if you do not owe or are getting a refund.  There is no place to enter on your tax return that you have an extension.    You will be able to e-file up until October 15, and then e-filing will close down.