I have come across a petition for naturalization that appears to be a request by parents for a child. Do I do a separate record for the child, father, and mother?
No, you only index the information for the child. You can find the instruction under What To Remember About this Project:
I came across the same type of image, but the child's name (Teresa McMahon) was different than what they wanted her to be known by. Another place on the image shows the child's name when she first came to U.S ( Sara Jane Lastname). Do I include both given names and surnames?
When you have a variation of a name, you index both names with the word or between them:
Given Name: Teresa or Sara Jane
Surname: McMahon or Lastname
For more on this you can read about Alias Names in the General Indexing Guidelines.