What do you mean by "the software I purchased"? The tag below your question indicates that you are using TurboTax Online. That is not software that you purchased, even if you paid for it. A TurboTax Online account can be used for only one person (or one married couple filing jointly). To do a tax return for another person you have to use a different account or create a new account.
But if you purchased TurboTax desktop software on a CD or as a download, and installed it on your computer, you can do as many tax returns as you want to. On the first screen when you start the TurboTax program, click the button that says "Start a New Return" (not Add a Person). To get back to that screen, select "New Tax Return" on the File menu (not the File tab).
What do you mean "Add a person"?
Are you trying to add a dependent?
What did you purchase? did you pay for the Online version or buy the Desktop CD/Download program?
Be sure you are in your Desktop program. People who bought Desktop have accidentally got into the Online version. They are different. Did you install and Activate the program with the 16 digit License Code? If you started filling out the Online version we can tell you how to move it to the Desktop program without paying for Online.
The Online version has a black column on the left side with the menu items, Federal, State, File etc.
The Windows Desktop program is blue across the top and the menu items like Federal, State, Review, File go across the top.