If your hands were completely dry, your chances are pretty minimal. In the future, avoid going b/w his genitals and your own, use latex gloves, or wash your hands w/ warm water and soap after touching him/before touching yourself.
I'd also suggest keeping better track of your periods. 'The first few days of June' isn't overly helpful ... You could be up to a week late (or not that late at all) and we wouldn't know the difference b/c you don't know the date of your last period. I'm not at all organized when it comes to keeping track of stuff either, but i put mine on a website ... http://www.cyclespage.com ... Love it. It keeps track of it all for me and all i have to do is remember to put in the first day of my period each month.
You can take a pregnancy test now that your period is late. We usually tell people 10 days after the last risk or the first day of their missed period.
I'm not sure if you can visit the doctor w/o your parents knowing b/c i do'nt know where you live ... But i'm going to assume that you can, provided that you're paying for the visit yourself (if you use your parents insurance, they may find out). If you're unsure, you can always call the office first and ask. Do you have a Planned Parenthood in your area? I've heard that many clinics have fees on a sliding scale (meaning you won't have to use your parents insurance, and it may be more affordable) ... Again, call and ask.
Why do you feel you have to 'dance around' this w/ your parents? Even assuming they don't know that you're sexually active, surely they can assume you're menstruating by now ...? And all menstruating women require annual GYN exams. (And menstruating or not, most doctors suggest annual physicals anyway ... so ... ) While we don't ever recommend lying to parents, that is ultimately up to you. You parents may suprise you tho.