I had never heard of wild parsnip, had to google it. Interesting that it is literally a burn brought on by UV exposure after getting the juice on your skin.
Is it possible you may have gotten the juice on your skin when removing the protective clothing? I have been known to get a poison ivy rash from touching the clothing of someone who had been working around poison ivy, and even from petting a cat after it had brushed past poison ivy.
I'm betting you are fair-skinned. The article I read said fair-skinned people can be extra-sensitive to tiny amounts of juice. (Fair-skinned people have more skin problems in general.) http://dnr.wi.gov/wnrmag/html/stories/1999/jun99/parsnip.htm
Doxycycline is often used for acne and for rosacea (which I have) so I'm not surprised that it helps you. Even though it's an antibiotic, it has a known anti-inflammatory effect.