I mulch parsnips with partially composted woodchip, but it's not yet the time (for me anyway). I'll wait until the plants are bigger so that I can safely apply 4 inches of material around them. Also, I'll mulch after heavy rain to keep the moisture in. Mulching dry soil is a stable door/horse bolted scenario.
Ben Acre is right tho'; parsnips' extremely deep roots will follow moisture down as the surface soil dries out so they are pretty good at resisting drought. That said, severe drought might reduce the eventual eating quality.