In Italian "pastinaca" and "dolce". I am not sure if parsnips are a common feature of Italian cuisine. I have never seen an Italian dish with it.
In the UK, "pudding" tends to be more often associated with hot desserts e.g. rice pudding, sticky toffee pudding, plum pudding, Christmas pudding. But it is also an exact synonym of "dessert".
Parsnip is a vegetable
Pudding is another English word for dessert