At one point you say "reading between the lines" and at another point you say "reading through the lines." The first one is more correct, but since this is more of an academic essay, I'd recommend using a less idiomatic word, if you can. In your thesis, maybe strike "reading between the lines" altogether.
a year after Billy Joe's suicidal
should be "suicide"
still considered "child" by parents.
Try not to use quotation marks unless what's between them is actually in the text.
The speaker indeed have intimate relationship with Billy Joe.
"has" not "have." Or, put a "does" after "speaker."
I assume that they has had sexual relationship at some point
They "have," not "has."
due to father's antagonism toward Billy
Whose father? Or is it just Father?
the speaker "haven't touch a single bite" of the pie that her mother has been "cooking all morning".
The speaker "[doesn't] touch a single bite" of the pie that her mother had been "cooking all morning." Since this is an academic paper, try to keep everything in the present tense.
the guilt from killing his own child triggered Billy Joe to terminate
present tense: triggers
which leave audience with unsolved enigmas
which leaves the audience with unsolved enigmas.
Good luck! =)