Q:

influence or influences?

"Parents play a crucial role in determining what does, or doesn't, influence their teens."

In the above sentence, should the word "influence" be replaced by "influences"? Please explain.

A:

No.  With a bit of reordering for grammatical clarity, it becomes obvious that the verb phrase in its entirety is "does or does not influence".  In this case, because the verb phrase is "does influence" the issue of subject-verb agreement is handled by the "does".  
For example: 

It does influence their teens.
They do influence their teens.  

