Placing a Parsnip in the crafting grid gives you a Parsnip Seed.
To farm the Parsnip, you plant and grow the seeds (or the Parsnip itself) on tilled soil, once they're fully grown you can right-click them, harvest the Parsnip and let them grow again. When harvested by right-click, the plant doesn't break, and yields 1-3 Parsnip.
Shapeless Crafting Edit
Here is a list of all recipes that use Parsnip: