Sorry to hear about your situation. I myself have been in that spot only once, but the circumstances were a bit different, and it was bout 4 years ago..... it was a very LTR relationship (won't go into detail), and I caught her cheating... A month after I kicked her out, I got a nasty message on my answering machine saying she was pregnant, and it was 'probably' mine...... went to talk with her about it, and she said no, it is 100% for sure mine..... I kept calling BS, saying the message specifically said PROBABLY.... asked her if there was a one in a trillion shot that it wouldn't be mine, and there was, but it was only once, and they used a condom, so it had to have been mine. Turned out when she gave me all the dates for the due date and everything, the timeline just wasn't agreeing that it was mine. It turned out that it wasn't mine.
Here is what I suggest:
FIRST: Go to the doctor with her, and YOU ask the Doctor questions about how far along she is, what the due date is, etc. etc. etc. They will be talking in terms of weeks typically, not months or days, and keep in mind that the human gestation period is 9 months. If she says that she will go before you get back, tell her that is fine, but she should plan on making an ADDITIONAL trip when you return. Do not take 'he said she said', or paperwork (unless you are able to call the office to confirm things, but you likely won't, Doctor patient confidentiality clauses and what not.... it would be different if you are there in person).
SECONDLY: No IFS, ANDS, or BUTS, about it, you NEED to have a paternity test done. While YES the ones that are legally binding in court are expensive (Mine was $500 I paid in cash), you CAN get take home ones that you mail in yourself (these will NOT hold up in court though). You should tell her that you want the test done, and you would like for her to pay for half of the testing fee. I have a friend who used this to prove the paternaty of her youngest son, and it was positive (she had only been with 1 guy, so wasn't rocket science).... I don't have any of the info about it, but I will see if I can get some from her when I see her Wednesday, and I will send you a PM or something (unless you find something yourself before that).
As stated many times, The money for paternaty test is FAR less than that of what you will end up paying in child support.... not to mention that you can be arrested for skipping your child support.
As far as your parents go, they are all different. I had told mine about my issue (even though it was an ENTIRELY different situation), and they backed me. It comes down to this, you are their SON, and they should love you un-conditionally, mistakes and all. If you tell them, and they have some kind of big issue with that and try to dis-own you or something, I would have to imagine that they must be perfect people, who live without any kind of mistakes what-so-ever (I personally would probably let them know that).
Good luck man! Even if it does end up being your kid, the best thing you can do is go into it with an open mind. Don't worry about the girl if a relationship doesn't work out, but you should still make an effort to be a part of your child's life.