Yes you can go to the ER but am not sure how long you'd have to wait. And the advantage of that would be easy referral to other places if you need

in our country they have sexual health clinics that will do a full STD screen confidentially, they even let you use a fake name - you just have to remember what name you used when you go back!!

But depending on how old you are, even if you go to your doctor they should not be able to tell your parents just because you wanted a test.