Ah, I just wanted to make sure it was actually parsnip since I haven't seen much lately. Treat it like any burn - a cool wet cloth will help, don't pop the blisters, cortizone steroids can help.
The technu relief might help (as would any anti itch/analgesic) but the cleanser (most common) won't do anything because it is made to break down the oils in poison ivy and parsnip doesn't have the oil based sap that poison ivy does.
Parsnip won't spread unless you're spreading the sap on your skin AND you get sun so don't worry too much about it spreading. The sap from broken leaves and stems makes you more sensitive to the sun... photophyto dermititis I think is the official name. You could rub the plant all over your body and sit inside and you'd never get a rash. I usually know when I've been in it and I've found if you wash with water only it will spread the sap around so you get a worse sunburn.. still much better than the big blisters. Wash with soap AND water if you can but I know that isn't always possible in the field.
Good luck!